HLIBMathematics

Mathematics: Analysis and Approaches

The Higher Level pathway extends the AA SL foundations through advanced algebra, functions, geometry, probability and calculus.

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Course Overview

IB Mathematics: Analysis and Approaches HL is organized into 5 units and 83 focused topics with 1311 Key Concepts sections, 1048 questions, and 1660 flashcards.

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Begin with Topic 1.1: Scientific notation and operations with numbers.

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Course units

Each unit groups focused topics. Every topic follows the same clear rhythm: review Key Concepts, use the Question Bank, then strengthen recall.

Unit 1
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Number and Algebra
AA HL includes the common SL foundations in number and algebra, then extends them through counting, generalized binomial expansions, partial fractions, complex numbers, advanced proof and linear systems.
16 topics
Unit 1 progress0%
Topics in this unit
Topic 1.1Not started
Scientific notation and operations with numbers
Scientific notation is a compact, reliable way to represent, compare and calculate with extremely large or small numbers. This guide builds from place value and conversions to arithmetic, significant figures, calculator use and IB-style multi-step problems.
105 min estimated
Topic 1.1 progress0%
Key Concepts0/8 sections complete Question Bank0/10 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Write every non-zero number in normalized scientific notation a × $10^{k}$ , where 1 ≤ |a| < 10 and k is an integer.
Convert accurately between ordinary decimal notation, scientific notation and calculator E-notation.
Multiply, divide and raise scientific-notation values to integer powers using exponent laws.
Add and subtract scientific-notation values by first expressing them with a common power of ten.
Round values to a stated number of significant figures and communicate trailing significant zeros clearly.
Use a scientific calculator efficiently while retaining guard digits until the final answer.
Estimate orders of magnitude and use them to check whether a calculated result is reasonable.
Present IB exam solutions with correct working, notation, accuracy and units.
Key formulas
Normalized scientific notation
$N=a\times10^k,\quad 1\le |a|<10,\quad k\in\mathbb Z$
Multiplication rule
$(a\times10^m)(b\times10^n)=(ab)\times10^{m+n}$
Division rule
$\frac{a\times10^m}{b\times10^n}=\frac{a}{b}\times10^{m-n},\quad b\ne0$
Power rule
$(a\times10^m)^p=a^p\times10^{mp}$
Addition with a common exponent
$a\times10^k+b\times10^k=(a+b)\times10^k$
Subtraction with a common exponent
$a\times10^k-b\times10^k=(a-b)\times10^k$
Calculator E-notation
$a\text{E}k=a\times10^k$
Topic 1.2Not started
Arithmetic sequences and series
Arithmetic sequences model quantities that change by the same additive amount each step. This guide develops common differences, nth terms, finite sums, sigma notation and IB-style applications from first principles.
110 min estimated
Topic 1.2 progress0%
Key Concepts0/8 sections complete Question Bank0/14 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Recognize an arithmetic sequence and calculate its common difference from consecutive or non-consecutive terms.
Use recurrence notation and the nth-term formula to generate and find terms.
Determine an unknown first term, common difference, term number or number of terms from given information.
Distinguish an arithmetic sequence from an arithmetic series and calculate the sum of a finite number of terms.
Represent and evaluate finite arithmetic sums using sigma notation.
Build and interpret arithmetic models for real-world situations with constant additive change.
Use calculator tables, summation tools and equation solvers efficiently while checking integer constraints.
Present IB exam solutions with clear formulas, substitutions, units and reasonableness checks.
Key formulas
Common difference
$d=u_{n+1}-u_n$
Recursive definition
$u_1=a,\qquad u_{n+1}=u_n+d$
Arithmetic nth term
$u_n=u_1+(n-1)d$
Common difference from two terms
$d=\frac{u_q-u_p}{q-p},\qquad p\ne q$
Term number
$n=1+\frac{u_n-u_1}{d},\qquad d\ne0$
Arithmetic series using first and last terms
$S_n=\frac n2(u_1+u_n)$
Arithmetic series using the common difference
$S_n=\frac n2\left[2u_1+(n-1)d\right]$
Arithmetic series in sigma notation
$S_n=\sum\limits_{r=1}^{n}\left[u_1+(r-1)d\right]$
Equivalent zero-based indexing
$S_n=\sum\limits_{r=0}^{n-1}(u_1+rd)$
Topic 1.3Not started
Geometric sequences and series
Geometric sequences change by a constant ratio. AA HL questions may connect geometric growth or decay to repeated percentage change, modelling and series notation.
120 min estimated
Topic 1.3 progress0%
Key Concepts0/9 sections complete Question Bank0/8 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Identify the common ratio and construct terms.
Use the nth-term formula and finite geometric sum.
Recognize when a geometric model is appropriate in context.
Key formulas
Common ratio
$r=\frac{u_{n+1}}{u_n},\qquad u_n\ne0$
Recursive definition
$u_1=a,\qquad u_{n+1}=ru_n$
Geometric nth term
$u_n=u_1r^{n-1}$
Ratio from two terms
$r^{q-p}=\frac{u_q}{u_p}$
Term number for a positive ratio
$n=1+\frac{\ln(\frac{u_n}{u_1})}{\ln r},\qquad r>0,\ r\ne1,\quad u_1\ne0,\quad \frac{u_n}{u_1}>0$
Finite geometric sum
$S_n=\frac{u_1(1-r^n)}{1-r},\qquad r\ne1$
Equivalent finite-sum form
$S_n=\frac{u_1(r^n-1)}{r-1},\qquad r\ne1$
Special case r = 1
$S_n=nu_1$
Infinite geometric sum
$S_\infty=\frac{u_1}{1-r},\qquad |r|<1$
Remainder after n terms
$R_n=S_\infty-S_n=\frac{u_1r^n}{1-r},\qquad |r|<1$
Finite geometric series in sigma notation
$S_n=\sum\limits_{k=0}^{n-1}u_1r^k$
Infinite geometric series in sigma notation
$\sum\limits_{k=0}^{\infty}u_1r^k=\frac{u_1}{1-r},\qquad |r|<1$
Repeated growth or decay
$Q_t=Q_0r^t$
Topic 1.4Not started
Financial applications (current syllabus)
Financial mathematics applies exponential growth and decay to money over repeated periods. Students should understand how percentage rate, number of compounding periods and time units interact.
140 min estimated
Topic 1.4 progress0%
Key Concepts0/9 sections complete Question Bank0/4 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Model compound interest and repeated percentage growth.
Model depreciation and interpret the value after a given number of periods.
Translate annual, monthly or other periodic information consistently before substituting.
Key formulas
Simple interest earned
$I=Prt$
Simple-interest balance
$A=P(1+rt)$
Percentage multiplier
$\text{growth: }1+\frac{p}{100},\qquad \text{decay: }1-\frac{p}{100}$
Compound future value
$FV=PV(1+i)^n$
Periodic rate from two values
$i=\left(\frac{FV}{PV}\right)^{\frac{1}{n}}-1,\quad n>0,\ PV\ne0,\frac{\ FV}{PV}>0$
Number of periods
$\begin{gathered}n=\frac{\ln(FV/PV)}{\ln(1+i)}\\ PV\ne0,\quad FV/PV>0,\quad i>-1,\ i\ne0\end{gathered}$
Nominal-rate compound interest
$FV=PV\left(1+\frac{r}{m}\right)^{mt}$
Periodic rate and period count
$i=\frac{r}{m},\qquad n=mt$
Effective annual rate
$EAR=\left(1+\frac{r}{m}\right)^m-1$
Present value of one future amount
$PV=\frac{FV}{(1+i)^n}=FV(1+i)^{-n}$
Appreciation
$V_n=V_0(1+a)^n$
Reducing-balance depreciation
$V_n=V_0(1-d)^n$
Future value of an ordinary annuity
$FV=\begin{cases}R\frac{(1+i)^n-1}{i},&i\ne0\\ nR,&i=0\end{cases}$
Future value of an annuity due
$FV_{\mathrm{due}}=\begin{cases}R\frac{(1+i)^n-1}{i}(1+i),&i\ne0\\ nR,&i=0\end{cases}$
Deposit required for a target
$R=\begin{cases}FV\frac{i}{(1+i)^n-1},&i\ne0\\ FV/n,&i=0\end{cases}$
Present value of loan repayments
$PV=\begin{cases}R\frac{1-(1+i)^{-n}}{i},&i\ne0\\ nR,&i=0\end{cases}$
Periodic loan repayment
$R=\begin{cases}PV\frac{i}{1-(1+i)^{-n}},&i\ne0\\ PV/n,&i=0\end{cases}$
Outstanding balance after k payments
$B_k=\begin{cases}PV(1+i)^k-R\frac{(1+i)^k-1}{i},&i\ne0\\ PV-kR,&i=0\end{cases}$
Total interest on equal repayments
$I_{\mathrm{total}}=Rn-PV$
Topic 1.5Not started
Integer exponents and introduction to logarithms
This section links exponent rules to logarithms and introduces logarithms as inverse operations. Technology can be used for numerical evaluation, but students should understand what a logarithm represents.
125 min estimated
Topic 1.5 progress0%
Key Concepts0/9 sections complete Question Bank0/8 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Apply the laws of exponents for integer powers.
Evaluate common logarithms and natural logarithms numerically.
Connect exponential equations with logarithmic notation.
Key formulas
Positive integer exponent
$a^n=\underbrace{a\cdot a\cdots a}_{n\text{ factors}},\quad n\in\mathbb N$
Zero exponent
$a^0=1,\quad a\ne0$
Negative integer exponent
$a^{-n}=\frac{1}{a^n},\quad a\ne0$
Product law
$a^m a^n=a^{m+n}$
Quotient law
$\frac{a^m}{a^n}=a^{m-n},\quad a\ne0$
Power of a power
$(a^m)^n=a^{mn}$
Power of a product
$(ab)^n=a^n b^n$
Power of a quotient
$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n},\quad b\ne0$
Moving a negative power across a fraction bar
$\frac{a^{-m}}{b^{-n}}=\frac{b^n}{a^m},\quad a,b\ne0$
Exponential-logarithmic equivalence
$a^c=b\quad\Longleftrightarrow\quad\log_a b=c$
Common and natural logarithms
$\log x=\log_{10}x,\qquad\ln x=\log_e x$
Inverse identities
$\log_a(a^x)=x,\qquad a^{\log_a x}=x\quad(x>0)$
Product property
$\log_a(xy)=\log_a x+\log_a y$
Quotient property
$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$
Power property
$\log_a(x^k)=k\log_a x$
Useful exact values
$\log_a1=0,\qquad\log_a a=1$
Common-base principle
$a^{f(x)}=a^{g(x)}\Longrightarrow f(x)=g(x),\quad a>0,\ a\ne1$
Solving a basic exponential equation
$a^x=b\Longrightarrow x=\frac{\ln b}{\ln a},\quad a>0,\ a\ne1,\ b>0$
Single logarithm equation
$\log_a f(x)=c\Longleftrightarrow f(x)=a^c,\quad f(x)>0$
Equality of logarithms
$\log_a f(x)=\log_a g(x)\Longrightarrow f(x)=g(x),\quad f(x),g(x)>0$
Change-of-base formula
$\log_a b=\frac{\log b}{\log a}=\frac{\ln b}{\ln a}$
Topic 1.6Not started
Simple deductive proof
AA HL expects students to justify mathematical statements rather than rely only on numerical evidence. Proof questions are usually short, but notation and logical flow matter.
115 min estimated
Topic 1.6 progress0%
Key Concepts0/9 sections complete Question Bank0/15 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Build a numerical or algebraic argument from accepted facts.
Show a clear chain of equivalences or implications.
Distinguish an equation that is true for particular values from an identity that is true throughout its domain.
Key formulas
Implication structure
$P\Rightarrow Q$
Universal statement
$\forall n\in S,\quad P(n)$
Even integer
$n=2k,\quad k\in\mathbb Z$
Odd integer
$n=2k+1,\quad k\in\mathbb Z$
Divisibility
$d\mid n\quad\Longleftrightarrow\quad n=dk\text{ for some }k\in\mathbb Z$
Consecutive integers
$n,\ n+1,\ n+2,\ldots\quad(n\in\mathbb Z)$
Direct-proof chain
$P\ \Longrightarrow\ R_1\ \Longrightarrow\ R_2\ \Longrightarrow\cdots\Longrightarrow\ Q$
Identity notation
$f(x)\equiv g(x)$
Difference of squares
$A^2-B^2=(A-B)(A+B)$
Square is non-negative
$u^2\ge0\quad\text{for every }u\in\mathbb R$
Even target form
$E=2K,\quad K\in\mathbb Z$
Odd target form
$O=2K+1,\quad K\in\mathbb Z$
Odd-square result
$n\text{ odd}\quad\Longrightarrow\quad n^2\equiv1\pmod8$
Divisibility target
$d\mid N\quad\Longleftrightarrow\quad N=dK\text{ for some }K\in\mathbb Z$
Linear-combination property
$d\mid a\ \land\ d\mid b\quad\Longrightarrow\quad d\mid(ra+sb),\quad r,s\in\mathbb Z$
Consecutive integers
$n,\ n+1,\ n+2$
Consecutive even integers
$2k,\ 2k+2,\ 2k+4$
Consecutive odd integers
$2k+1,\ 2k+3,\ 2k+5$
Counterexample test
$P(c)\text{ true and }Q(c)\text{ false}\quad\Longrightarrow\quad \forall x\,[P(x)\Rightarrow Q(x)]\text{ is false}$
Topic 1.7Not started
Rational exponents and logarithm laws
Students extend exponent rules to rational powers and use logarithm laws to simplify expressions and solve exponential equations. This topic appears frequently inside function and calculus questions as well.
125 min estimated
Topic 1.7 progress0%
Key Concepts0/9 sections complete Question Bank0/32 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Apply product, quotient and power laws for logarithms.
Change the base of a logarithm when necessary.
Solve exponential equations by taking logarithms and check domain restrictions.
Key formulas
Rational exponent definition
$a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$
Negative rational exponent
$a^{\frac{-m}{n}}=\frac{1}{a^{\frac{m}{n}}}$
Special root form
$a^{\frac{1}{n}}=\sqrt[n]{a}$
Radical–exponent conversion
$\sqrt[n]{x^m}=x^{\frac{m}{n}},\qquad x>0,\ m\in\mathbb Z,\ n\in\mathbb Z_{>0}$
Principal square root
$\sqrt{x^2}=|x|$
Odd root of a power
$\sqrt[2k+1]{x^{2k+1}}=x$
Product law
$a^r a^s=a^{r+s}$
Quotient law
$\frac{a^r}{a^s}=a^{r-s}$
Power of a power
$(a^r)^s=a^{rs}$
Power of a product
$(ab)^r=a^r b^r$
Power of a quotient
$\left(\frac ab\right)^r=\frac{a^r}{b^r}$
Negative exponent conversion
$x^{-r}=\frac{1}{x^r}$
Common-denominator strategy
$x^{\frac{p}{q}}x^{\frac{r}{s}}=x^{\frac{(ps+rq)}{(qs)}}$
Exact before decimal
$\sqrt[n]{a^m}=a^{\frac{m}{n}}$
Logarithm definition
$\log_a x=y\quad\Longleftrightarrow\quad a^y=x$
Logarithm of one
$\log_a1=0$
Logarithm of its base
$\log_a a=1$
Inverse relationships
$a^{\log_a x}=x,\qquad \log_a(a^x)=x$
Product law
$\log_a(xy)=\log_a x+\log_a y$
Quotient law
$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$
Power law
$\log_a(x^r)=r\log_a x$
Reciprocal result
$\log_a\left(\frac1x\right)=-\log_a x$
Change of base
$\log_a x=\frac{\log_b x}{\log_b a}$
Calculator forms
$\log_a x=\frac{\ln x}{\ln a}=\frac{\log x}{\log a}$
Condensing pattern
$r\log_a x-s\log_a y=\log_a\left(\frac{x^r}{y^s}\right)$
Same-base principle
$a^u=a^v\quad\Longrightarrow\quad u=v$
Logarithmic solution step
$a^{f(x)}=k\quad\Longrightarrow\quad f(x)=\frac{\ln k}{\ln a}$
Log equality principle
$\log_a u=\log_a v\quad\Longrightarrow\quad u=v$
Fractional exponent
$a^{\frac{m}{n}}=\sqrt[n]{a^m}$
Three log laws
$\log_a(xy)=\log_ax+\log_ay,\quad \log_a(\frac{x}{y})=\log_ax-\log_ay,\quad \log_a(x^r)=r\log_ax$
Change of base
$\log_a x=\frac{\ln x}{\ln a}$
Topic 1.8Not started
Infinite geometric series
An infinite geometric series has a finite sum only when its terms shrink fast enough. Students must identify convergence before applying the sum formula.
110 min estimated
Topic 1.8 progress0%
Key Concepts0/9 sections complete Question Bank0/11 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Determine whether |r| < 1.
Find the limiting sum of a convergent geometric series.
Use an infinite series model in an applied or algebraic context.
Key formulas
Infinite geometric series
$a+ar+ar^2+ar^3+\cdots$
Partial sum notation
$S_n=\sum\limits_{k=0}^{n-1}ar^k$
Common ratio
$r=\frac{u_{n+1}}{u_n}$
Geometric term
$u_n=ar^{n-1}$
Geometric convergence condition
$|r|<1\quad\Longleftrightarrow\quad -1<r<1$
Vanishing powers
$|r|<1\quad\Longrightarrow\quad r^n\to0$
Finite geometric sum
$S_n=\frac{a(1-r^n)}{1-r}$
Sum to infinity
$S_\infty=\frac{a}{1-r},\qquad |r|<1$
First term from the sum
$a=S_\infty(1-r)$
Ratio from first term and sum
$r=1-\frac{a}{S_\infty},\qquad S_\infty\ne0$
Term condition
$u_n=ar^{n-1}$
Remainder after n terms
$R_n=S_\infty-S_n=\frac{ar^n}{1-r}$
Approximation relation
$S_\infty=S_n+R_n$
Bouncing-ball distance
$D=h+\frac{2hq}{1-q}=h\left(\frac{1+q}{1-q}\right)$
Repeated fixed contribution
$A_\infty=\frac{d}{1-r},\qquad |r|<1$
Two-digit recurring decimal
$0.\overline{ab}=\frac{10a+b}{99},\qquad a,b\in\{0,1,\ldots,9\}$
Partial-sum calculator entry
$S_n=\frac{a(1-r^n)}{1-r}$
Solver interval
$-1<r<1$
Convergence and sum
$|r|<1\quad\Longrightarrow\quad S_\infty=\frac{a}{1-r}$
Term and remainder
$u_n=ar^{n-1},\qquad R_n=\frac{ar^n}{1-r}$
Topic 1.9Not started
Binomial theorem
The binomial theorem provides a systematic expansion of powers such as $\left(a + b\right)^{n.}$ Students should be able to identify individual terms without expanding the entire expression.
115 min estimated
Topic 1.9 progress0%
Key Concepts0/9 sections complete Question Bank0/43 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Use binomial coefficients and Pascal’s triangle.
Find a specified term or coefficient in an expansion.
Connect the general term with powers of the two binomial components.
Key formulas
Binomial theorem
$(a+b)^n=\sum\limits_{r=0}^{n}\binom nr a^{n-r}b^r$
Number of formal binomial summands
$\text{formal summands before collection}=n+1$
Pascal recurrence
$\binom nr=\binom{n-1}{r-1}+\binom{n-1}{r}$
Coefficient symmetry
$\binom nr=\binom n{n-r}$
Row sum
$\sum\limits_{r=0}^{n}\binom nr=2^n$
Combination formula
$\binom nr=\frac{n!}{r!(n-r)!}$
Factorial
$n!=n(n-1)(n-2)\cdots2\cdot1,\qquad 0!=1$
Boundary values
$\binom n0=\binom nn=1$
Expanded form
$(a+b)^n=\binom n0a^n+\binom n1a^{n-1}b+\cdots+\binom nnb^n$
Exponent check
$(n-r)+r=n$
General term
$T_{r+1}=\binom nr a^{n-r}b^r$
pth-term index
$r=p-1$
Coefficient method
$\text{power of }x=k\quad\Longrightarrow\quad \text{solve for }r$
Quadratic coefficient
$\binom n2=\frac{n(n-1)}2$
Constant-term condition
$\text{net exponent of }x=0$
Admissible index
$r\in\{0,1,2,\ldots,n\}$
Sum of coefficients
$\sum\limits_{k=0}^{n}c_k=P(1)$
Alternating coefficient sum
$\sum\limits_{k=0}^{n}(-1)^kc_k=P(-1)$
GDC term calculation
$\binom nr a^{n-r}b^r$
Core theorem
$(a+b)^n=\sum\limits_{r=0}^{n}\binom nr a^{n-r}b^r$
Target term
$T_{r+1}=\binom nr a^{n-r}b^r$
Constant condition
$\text{net power of }x=0$
Topic 1.10Not started
Counting principles and the extended binomial theorem
Higher Level extends counting beyond the coefficients used in ordinary binomial expansion. Students distinguish arrangements from selections, use factorial notation efficiently and apply generalized binomial expansions when the exponent is not a non-negative integer.
165 min estimated
Topic 1.10 progress0%
Key Concepts0/14 sections complete Question Bank0/16 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Use systematic counting principles to determine the number of possible outcomes without double-counting.
Distinguish permutations, where order matters, from combinations, where order does not matter.
Use factorial notation and calculate nPr and nCr values analytically or with technology where appropriate.
Use the generalized binomial expansion for rational, fractional or negative exponents and identify a required coefficient or term.
Recognize the interval/condition under which an infinite binomial expansion is valid.
Key formulas
Fundamental counting principle
$N=m_1m_2\cdots m_k$
Sum rule for disjoint cases
$N=N_1+N_2+\cdots+N_k$
Factorial definition
$n!=n(n-1)(n-2)\cdots2\cdot1$
Zero factorial
$0!=1$
Useful cancellation
$\frac{n!}{(n-r)!}=n(n-1)\cdots(n-r+1)$
Arrange all distinct objects
$N=n!$
Permutations of r from n
${}_{n}P_r=\frac{n!}{(n-r)!}$
Combination formula
$\binom{n}{r}={}_{n}C_r=\frac{n!}{r!(n-r)!}$
Combination symmetry
$\binom{n}{r}=\binom{n}{n-r}$
Permutation-combination link
${}_{n}P_r=r!\binom{n}{r}$
Arrangements with identical objects
$N=\frac{n!}{n_1!n_2!\cdots n_k!}$
Sequences with repetition allowed
$N=k^r$
Complement count
$N(\text{valid})=N(\text{all})-N(\text{forbidden})$
Finite binomial theorem
$(a+b)^n=\sum\limits_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$
Finite general term
$T_{r+1}=\binom{n}{r}a^{n-r}b^r$
Pascal identity
$\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}$
Generalized binomial coefficient
$\binom{\alpha}{r}=\frac{\alpha(\alpha-1)\cdots(\alpha-r+1)}{r!}$
Extended binomial theorem
$(1+x)^\alpha=\sum\limits_{r=0}^{\infty}\binom{\alpha}{r}x^r,\qquad |x|<1$
First four terms
$(1+x)^\alpha=1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+\frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3+\cdots$
Negative upper coefficient
$\binom{-m}{r}=(-1)^r\binom{m+r-1}{r}$
Normalized binomial
$(a+bx)^\alpha=a^\alpha\left(1+\frac{b}{a}x\right)^\alpha$
Resulting validity condition
$\left|\frac{b}{a}x\right|<1$
Extended general term
$T_{r+1}=\binom{\alpha}{r}u^r$
General term after normalization
$T_{r+1}=a^\alpha\binom{\alpha}{r}\left(\frac{b}{a}x\right)^r$
Standard convergence condition
$|x|<1$
Substituted convergence condition
$|g(x)|<1$
Truncated approximation
$(1+x)^\alpha\approx\sum\limits_{r=0}^{k}\binom{\alpha}{r}x^r$
Generalized coefficient recurrence
$\binom{\alpha}{r+1}=\binom{\alpha}{r}\frac{\alpha-r}{r+1}$
Plausibility bound for a restriction
$0\le N(\text{valid})\le N(\text{all})$
Counting decision
$\text{order matters}\Rightarrow {}_{n}P_r,\qquad \text{order does not matter}\Rightarrow\binom{n}{r}$
Extended-expansion checklist
$(a+bx)^\alpha=a^\alpha(1+u)^\alpha,\quad u=\frac{b}{a}x,\quad |u|<1$
Ordered selection
${}_{n}P_r=\frac{n!}{(n-r)!}$
Unordered selection
$\binom{n}{r}=\frac{n!}{r!(n-r)!}$
Factorial
$n!=n(n-1)\cdots2\cdot1,\qquad 0!=1$
Permutation and combination
${}_{n}P_r=\frac{n!}{(n-r)!},\qquad \binom{n}{r}=\frac{n!}{r!(n-r)!}$
Repeated objects
$N=\frac{n!}{n_1!n_2!\cdots n_k!}$
Extended binomial theorem
$(1+u)^\alpha=\sum\limits_{r=0}^{\infty}\binom{\alpha}{r}u^r,\qquad |u|<1$
Generalized coefficient
$\binom{\alpha}{r}=\frac{\alpha(\alpha-1)\cdots(\alpha-r+1)}{r!}$
Topic 1.11Not started
Partial fractions
Partial fractions reverse the process of combining rational expressions. The technique is important because a complicated rational function can often be rewritten as a sum of simpler fractions that is easier to integrate, simplify or analyze.
155 min estimated
Topic 1.11 progress0%
Key Concepts0/14 sections complete Question Bank0/10 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Check whether a rational expression is proper before decomposing it; use polynomial division first if necessary.
Factor the denominator and choose a decomposition that matches its factor structure.
Solve for unknown constants by equating coefficients or substituting convenient values.
Use the decomposition later in algebraic simplification and integration problems.
Key formulas
Basic decomposition identity
$\frac{P(x)}{Q(x)}=\sum\limits_{k=1}^{m}\frac{N_k(x)}{F_k(x)}$
Polynomial division form
$P(x)=Q(x)S(x)+R(x)$
Improper rational expression
$\frac{P(x)}{Q(x)}=S(x)+\frac{R(x)}{Q(x)}$
Remainder condition
$\operatorname{degree}R<\operatorname{degree}Q$
Repeated linear factor template
$\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots+\frac{A_m}{(x-a)^m}$
Irreducible quadratic template
$\frac{Bx+C}{ax^2+bx+c}$
Quadratic irreducibility test
$b^2-4ac<0$
Two distinct linear factors
$\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}$
Cover-up coefficient
$A=\left.\frac{P(x)}{\frac{Q(x)}{(x-a)}}\right|_{x=a}$
Repeated linear decomposition
$\frac{P(x)}{(x-a)^mR(x)}=\frac{A_1}{x-a}+\cdots+\frac{A_m}{(x-a)^m}+\text{terms from }R(x)$
Highest-power coefficient
$A_m=\left.\frac{P(x)}{R(x)}\right|_{x=a}$
Irreducible quadratic contribution
$\frac{Bx+C}{ax^2+bx+c}$
Repeated quadratic contribution
$\sum\limits_{j=1}^{m}\frac{B_jx+C_j}{q(x)^j}$
Coefficient at a simple root
$A=\lim_{x\to a}(x-a)\frac{P(x)}{Q(x)}$
Highest repeated coefficient
$A_m=\lim_{x\to a}(x-a)^m\frac{P(x)}{Q(x)}$
Polynomial coefficient identity
$a_nx^n+\cdots+a_0=b_nx^n+\cdots+b_0\Rightarrow a_k=b_k$
Reconstruction check
$\sum\limits_{k=1}^{m}\frac{N_k(x)}{F_k(x)}=\frac{\frac{\sum\limits_{k=1}^{m}N_k(x)\,Q(x)}{F_k(x)}}{Q(x)}$
Linear-factor integral
$\int\frac{A}{x-a}\,dx=A\ln|x-a|+C$
Basic telescoping decomposition
$\frac1{n(n+1)}=\frac1n-\frac1{n+1}$
Complete workflow
$\text{divide}\to\text{factor}\to\text{template}\to\text{clear}\to\text{solve}\to\text{verify}$
Reasonableness identity
$P(x)\stackrel{?}{=}\sum\limits_{k=1}^{m}N_k(x)\frac{Q(x)}{F_k(x)}$
Distinct linear template
$\frac{A}{x-a}+\frac{B}{x-b}$
Repeated linear template
$\sum\limits_{j=1}^{m}\frac{A_j}{(x-a)^j}$
Quadratic template
$\frac{Bx+C}{ax^2+bx+c}$
Improper-expression step
$\frac{P}{Q}=S+\frac{R}{Q},\qquad \operatorname{degree}R<\operatorname{degree}Q$
Distinct linear factors
$\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}$
Repeated linear factor
$\sum\limits_{j=1}^{m}\frac{A_j}{(x-a)^j}$
Repeated irreducible quadratic
$\sum\limits_{j=1}^{m}\frac{B_jx+C_j}{q(x)^j}$
Topic 1.12Not started
Complex numbers in Cartesian form
Complex numbers extend the real number system so that equations such as $x^{2} + 1 = 0$ have solutions. Students work algebraically with a + bi form and geometrically with points and vectors in the Argand plane.
160 min estimated
Topic 1.12 progress0%
Key Concepts0/15 sections complete Question Bank0/12 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Use $i^{2} = -1$ to simplify powers of i and arithmetic involving complex numbers.
Identify the real part, imaginary part, conjugate, modulus and argument of a complex number.
Add, subtract, multiply and divide complex numbers in Cartesian form.
Represent complex numbers on an Argand diagram and interpret modulus as distance from the origin.
Use the conjugate to simplify quotients and connect complex roots with polynomial equations.
Key formulas
Imaginary unit
$i^2=-1$
Cartesian form
$z=a+bi$
Set of complex numbers
$\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}$
Real part
$\operatorname{Re}(a+bi)=a$
Imaginary part
$\operatorname{Im}(a+bi)=b$
Real-number condition
$z\in\mathbb{R}\iff\operatorname{Im}(z)=0$
Complex equality
$a+bi=c+di\iff a=c\text{ and }b=d$
Addition
$(a+bi)+(c+di)=(a+c)+(b+d)i$
Subtraction
$(a+bi)-(c+di)=(a-c)+(b-d)i$
Four-power cycle
$i^0=1,\quad i^1=i,\quad i^2=-1,\quad i^3=-i,\quad i^4=1$
Periodicity
$i^{n+4}=i^n$
Inverse of i
$i^{-1}=-i$
Complex multiplication
$(a+bi)(c+di)=(ac-bd)+(ad+bc)i$
Complex square
$(a+bi)^2=(a^2-b^2)+2abi$
Conjugate
$\overline{a+bi}=a-bi$
Sum with conjugate
$z+\overline z=2\operatorname{Re}(z)$
Difference from conjugate
$z-\overline z=2i\operatorname{Im}(z)$
Conjugate product
$z\overline z=a^2+b^2$
Conjugate of a product
$\overline{zw}=\overline z\,\overline w$
Complex quotient
$\frac{a+bi}{c+di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$
Complex reciprocal
$\frac1{a+bi}=\frac{a-bi}{a^2+b^2}$
Modulus
$|a+bi|=\sqrt{a^2+b^2}$
Modulus squared
$|z|^2=z\overline z$
Distance
$d(z_1,z_2)=|z_1-z_2|$
Product modulus
$|zw|=|z||w|$
Triangle inequality
$|z+w|\le |z|+|w|$
Argand coordinate
$z=a+bi\longleftrightarrow(a,b)$
Conjugate reflection
$a+bi\longleftrightarrow a-bi$
Displacement
$z_1-z_2$
Square in Cartesian form
$(a+bi)^2=(a^2-b^2)+2abi$
Quadratic formula
$z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Conjugate factor pair
$(x-(a+bi))(x-(a-bi))=(x-a)^2+b^2$
Quadratic from conjugate roots
$x^2-2ax+(a^2+b^2)=0$
Circle locus
$|z-c|=r$
Equidistance locus
$|z-z_1|=|z-z_2|$
Universal arithmetic check
$\text{computed answer}\xrightarrow{\text{substitution}}\text{original equation}$
Core Cartesian workflow
$\text{expand}\to\text{use }i^2=-1\to\text{collect real and imaginary parts}$
Conjugate
$\overline z=a-bi$
Modulus
$|z|=\sqrt{a^2+b^2}$
Division
$\frac{a+bi}{c+di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$
Topic 1.13Not started
Polar, cis and exponential forms of complex numbers
Polar form separates a complex number into its modulus and direction. It is especially efficient for multiplication, division, powers and roots, where geometric rotation and scaling become visible.
160 min estimated
Topic 1.13 progress0%
Key Concepts0/16 sections complete Question Bank0/5 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Convert between Cartesian and polar/exponential forms.
Choose an argument consistent with the point’s quadrant and the required argument convention.
Multiply and divide complex numbers by combining moduli and arguments.
Interpret multiplication geometrically as a scaling and rotation in the complex plane.
Key formulas
Modulus
$r=|z|=\sqrt{x^2+y^2}$
Cartesian projections
$x=r\cos\theta,\qquad y=r\sin\theta$
Tangent relation
$\tan\theta=\frac{y}{x}$
Polar form
$z=r(\cos\theta+i\sin\theta)$
Equivalent arguments
$\theta_k=\theta+2k\pi,\qquad k\in\mathbb Z$
Cartesian reconstruction
$z=r\cos\theta+i r\sin\theta$
Cis definition
$\operatorname{cis}\theta=\cos\theta+i\sin\theta$
Cis form
$z=r\operatorname{cis}\theta$
Cis periodicity
$\operatorname{cis}(\theta+2k\pi)=\operatorname{cis}\theta$
Euler's relation
$e^{i\theta}=\cos\theta+i\sin\theta$
Exponential form
$z=re^{i\theta}$
Equivalent notation
$re^{i\theta}=r\operatorname{cis}\theta$
Principal-argument convention
$-\pi<\operatorname{Arg}z\le\pi$
General arguments
$\arg z=\operatorname{Arg}z+2k\pi,\qquad k\in\mathbb Z$
Cartesian-to-polar modulus
$r=\sqrt{x^2+y^2}$
Cartesian-to-polar argument
$\theta=\operatorname{atan2}(y,x)$
Equivalent output forms
$r(\cos\theta+i\sin\theta)=r\operatorname{cis}\theta=re^{i\theta}$
Real component
$x=r\cos\theta$
Imaginary component
$y=r\sin\theta$
Polar-to-Cartesian conversion
$r\operatorname{cis}\theta=r\cos\theta+i r\sin\theta$
Product in cis form
$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$
Product in exponential form
$z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$
Product modulus
$|z_1z_2|=|z_1||z_2|$
Quotient in cis form
$\frac{z_1}{z_2}=\frac{r_1}{r_2}\operatorname{cis}(\theta_1-\theta_2)$
Reciprocal
$\frac1{r\operatorname{cis}\theta}=\frac1r\operatorname{cis}(-\theta)$
Conjugate in polar form
$\overline{r\operatorname{cis}\theta}=r\operatorname{cis}(-\theta)$
Quotient modulus
$\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$
Scaling and rotation
$z\mapsto k\operatorname{cis}\phi\,z$
Modulus transformation
$|k\operatorname{cis}\phi\,z|=k|z|$
Argument transformation
$\arg(k\operatorname{cis}\phi\,z)=\arg z+\phi\pmod{2\pi}$
Axis arguments
$\operatorname{Arg}(a)=0\ (a>0),\quad \operatorname{Arg}(a)=\pi\ (a<0)$
Imaginary-axis arguments
$\operatorname{Arg}(bi)=\frac{\pi}{2}\ (b>0),\quad \operatorname{Arg}(bi)=-\frac{\pi}{2}\ (b<0)$
Positive-modulus conversion
$-r\operatorname{cis}\theta=r\operatorname{cis}(\theta+\pi)$
Operation choice
$\text{add/subtract}\to\text{Cartesian},\qquad \text{multiply/divide}\to\text{polar}$
Quadrant-aware argument
$\theta=\operatorname{atan2}(y,x)$
Post-operation check
$|z_1z_2|=|z_1||z_2|,\qquad \arg(z_1z_2)=\arg z_1+\arg z_2\pmod{2\pi}$
Principal normalization
$\theta\mapsto\theta+2k\pi\in(-\pi,\pi]$
Conversion-operation workflow
$\text{identify form}\to\text{find }r,\theta\to\text{operate}\to\text{normalize}\to\text{verify}$
Modulus
$r=\sqrt{x^2+y^2}$
Product
$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$
Quotient
$\frac{z_1}{z_2}=\frac{r_1}{r_2}\operatorname{cis}(\theta_1-\theta_2)$
Topic 1.14Not started
Conjugate roots and De Moivre’s theorem
De Moivre’s theorem makes powers and roots of complex numbers systematic. Students also use the conjugate-root property of real-coefficient polynomials to connect complex solutions back to algebraic equations.
170 min estimated
Topic 1.14 progress0%
Key Concepts0/16 sections complete Question Bank0/10 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Use the fact that non-real roots of a polynomial with real coefficients occur in conjugate pairs.
Apply De Moivre’s theorem to calculate powers exactly.
Find all nth roots of a complex number and display their regular angular spacing on an Argand diagram.
Use polar form to solve equations involving powers of complex numbers.
Key formulas
Complex conjugate
$\overline{a+bi}=a-bi$
Conjugate product
$z\overline z=|z|^2=a^2+b^2$
Conjugate in polar form
$\overline{r\operatorname{cis}\theta}=r\operatorname{cis}(-\theta)$
Conjugation of a real polynomial
$p(\overline z)=\overline{p(z)}$
Conjugate-root implication
$p(z)=0\implies p(\overline z)=0$
Conjugate factor pair
$(x-(a+bi))(x-(a-bi))=(x-a)^2+b^2$
Expanded real quadratic
$x^2-2ax+(a^2+b^2)$
Repeated conjugate factor
$\left((x-a)^2+b^2\right)^m$
De Moivre's theorem
$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$
Unit-modulus form
$(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)$
Exponential equivalent
$(re^{i\theta})^n=r^ne^{in\theta}$
Power modulus
$|z^n|=|z|^n$
Power argument
$\arg(z^n)=n\arg z\pmod{2\pi}$
Triple-angle cosine
$\cos3\theta=4\cos^3\theta-3\cos\theta$
Triple-angle sine
$\sin3\theta=3\sin\theta-4\sin^3\theta$
Nth-root formula
$z_k=R^{\frac{1}{n}}\operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right),\qquad k=0,1,\ldots,n-1$
Root modulus
$|z_k|=R^{\frac{1}{n}}$
Root angular spacing
$\Delta\theta=\frac{2\pi}{n}$
Root angles
$\phi_k=\frac{\theta+2k\pi}{n}$
Root check
$(z_k)^n=w$
Root circle
$|z|=R^{\frac{1}{n}}$
Consecutive-root rotation
$z_{k+1}=z_k\operatorname{cis}\frac{2\pi}{n}$
Root sum
$\sum\limits_{k=0}^{n-1}z_k=0,\qquad n>1$
Nth roots of unity
$\omega_k=\operatorname{cis}\frac{2k\pi}{n},\qquad k=0,1,\ldots,n-1$
Root-of-unity equation
$z^n-1=0$
Primitive-root condition
$\gcd(k,n)=1$
Sum of nth roots of unity
$\sum\limits_{k=0}^{n-1}\omega_k=0,\qquad n>1$
Factor theorem
$p(\alpha)=0\iff(x-\alpha)\text{ is a factor of }p(x)$
Root-count rule
$\deg p=n\implies n\text{ complex roots counting multiplicity}$
Shifted power substitution
$(z-a)^n=w\quad\text{with }u=z-a$
Root-family completeness
$k=0,1,\ldots,n-1$
Numerical root check
$|p(z_k)|\approx0$
Geometric root check
$|z_k|=R^{\frac{1}{n}},\qquad \arg z_{k+1}-\arg z_k=\frac{2\pi}{n}$
Power checklist
$r\operatorname{cis}\theta\xrightarrow{n}r^n\operatorname{cis}(n\theta)$
Root checklist
$R\operatorname{cis}\theta\xrightarrow{n\text{th roots}}R^{\frac{1}{n}}\operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)$
Complete workflow
$\text{identify theorem}\to\text{polar form}\to\text{power or roots}\to\text{complete set}\to\text{verify}$
Conjugate-root factor
$(x-(a+bi))(x-(a-bi))=(x-a)^2+b^2$
Nth roots
$z_k=R^{\frac{1}{n}}\operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)$
Roots of unity
$\omega_k=\operatorname{cis}\frac{2k\pi}{n}$
Topic 1.15Not started
Proof by induction, contradiction and counterexample
Higher Level expects students to communicate proof with explicit logical structure. Induction establishes a statement for every integer in a sequence of cases; contradiction assumes the opposite and derives an impossibility; a counterexample disproves a universal statement by finding one valid exception.
175 min estimated
Topic 1.15 progress0%
Key Concepts0/16 sections complete Question Bank0/19 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Write a complete induction proof with a base case, inductive hypothesis and inductive step.
Use contradiction when assuming the negation of a statement leads to a logical impossibility.
For the current syllabus, use a counterexample to show that a universal claim is false.
Distinguish proof from numerical evidence: checking many cases is not the same as proving all cases.
Key formulas
Implication
$P\implies Q$
Universal statement
$\forall x\in D,\ P(x)$
Existential statement
$\exists x\in D\text{ such that }P(x)$
Contradiction form for an implication
$P\land\neg Q\implies\bot$
Disproving a universal claim
$\neg(\forall x\in D, P(x))\iff\exists x\in D\text{ with }\neg P(x)$
Induction principle
$P(n_0)\land\big(\forall k\ge n_0,\ P(k)\implies P(k+1)\big)\implies\forall n\ge n_0,\ P(n)$
Base-case instruction
$n=n_0:\quad\text{verify }P(n_0)$
Inductive hypothesis
$P(k)\text{ is assumed for an arbitrary }k\ge n_0$
Divisibility hypothesis
$d\mid A_k\iff A_k=dm\text{ for some }m\in\mathbb Z$
Sum extension
$\sum\limits_{r=1}^{k+1}a_r=\sum\limits_{r=1}^{k}a_r+a_{k+1}$
Inductive implication
$P(k)\implies P(k+1)$
Natural-number sum
$\sum\limits_{r=1}^{n}r=\frac{n(n+1)}2$
Odd-number sum
$\sum\limits_{r=1}^{n}(2r-1)=n^2$
Finite geometric sum
$\sum\limits_{r=0}^{n}q^r=\frac{q^{n+1}-1}{q-1}$
Divisibility definition
$d\mid A\iff A=dm\text{ for some }m\in\mathbb Z$
Useful power rearrangement
$a^{k+1}-1=a(a^k-1)+(a-1)$
One-step recurrence proof
$u_{k+1}=f(u_k)\xrightarrow{P(k)}f(g(k))=g(k+1)$
Strong induction hypothesis
$P(n_0),P(n_0+1),\ldots,P(k)$
Example inequality chain
$2^{k+1}=2\cdot2^k\ge2(k+1)\ge k+2$
Contradiction schema
$\neg S\implies\bot\quad\therefore\quad S$
Implication by contradiction
$P\land\neg Q\implies\bot\quad\therefore\quad P\implies Q$
Even integer
$n=2m,\qquad m\in\mathbb Z$
Odd integer
$n=2m+1,\qquad m\in\mathbb Z$
Euclid construction
$N=p_1p_2\cdots p_m+1$
Counterexample to implication
$P(c)\text{ true and }Q(c)\text{ false}$
Universal negation
$\neg(\forall x\in D,P(x))\iff\exists c\in D,\neg P(c)$
Induction checklist
$P(n_0);\quad P(k)\text{ assumed};\quad P(k+1)\text{ proved};\quad\text{conclude for all }n\ge n_0$
Contradiction checklist
$\text{assume negation}\to\text{deduce impossibility}\to\text{reject negation}$
Proof workflow
$\text{identify quantifier}\to\text{choose method}\to\text{justify every step}\to\text{state conclusion}$
Induction structure
$P(n_0);\quad P(k)\implies P(k+1);\quad\therefore\ P(n)\text{ for all }n\ge n_0$
Contradiction structure
$\neg S\implies\bot\quad\therefore\quad S$
Counterexample structure
$c\in D,\quad P(c)\text{ is false}\quad\therefore\quad\forall x\in D,P(x)\text{ is false}$
Topic 1.16Not started
Systems of linear equations
Students solve and interpret systems of two or three linear equations using algebra, matrices, Gaussian elimination and technology. They distinguish unique, inconsistent and dependent systems, including parameter cases, and connect row structure with the geometry of lines and planes.
175 min estimated
Topic 1.16 progress0%
Key Concepts0/17 sections complete Question Bank0/7 questions attempted Flashcards0/20 cards reviewed
Learning objectives
Explain a system of linear equations and verify an ordered pair or triple in every original equation.
Solve systems with two unknowns by substitution and elimination using exact arithmetic.
Solve systems with three unknowns by systematic elimination and back-substitution.
Represent a system as an augmented matrix and perform valid elementary row operations.
Use Gaussian elimination or reduced row-echelon form to solve and classify a system.
Distinguish unique solutions, no solutions and infinitely many solutions from equations, row-echelon form and geometry.
Identify consistent, inconsistent, independent and dependent systems and communicate free parameters correctly.
Analyze parameter-based systems without dividing by an expression that may be zero.
Use a GDC to solve or check systems while preserving exact setup, interpretation and validation.
Model real situations with linear systems, interpret the solution with units and test whether it is reasonable.
Key formulas
General two-variable linear system
$a_1x+b_1y=c_1,\qquad a_2x+b_2y=c_2$
General three-variable linear equation
$ax+by+cz=d$
Substitution pattern
$x=g(y)\quad\Longrightarrow\quad a\,g(y)+by=c$
Elimination combination
$\lambda(a_1x+b_1y=c_1)+\mu(a_2x+b_2y=c_2)$
Three-equation model
$\begin{aligned}a_1x+b_1y+c_1z&=d_1\\a_2x+b_2y+c_2z&=d_2\\a_3x+b_3y+c_3z&=d_3\end{aligned}$
Matrix form
$A\mathbf{x}=\mathbf{b}$
Augmented matrix
$[A\mid\mathbf{b}]$
Two-variable coefficient matrix
$A=\begin{bmatrix}a_1&b_1\\a_2&b_2\end{bmatrix}$
Row interchange
$R_i\leftrightarrow R_j$
Nonzero row scaling
$R_i\to\lambda R_i,\qquad\lambda\ne0$
Row replacement
$R_i\to R_i+\lambda R_j$
Row-echelon structure
$\begin{bmatrix}*&*&*\\0&*&*\\0&0&*\end{bmatrix}$
Back-substitution direction
$z\longrightarrow y\longrightarrow x$
Consistency criterion
$\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf b])$
Unique-solution rank condition
$\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf b])=n$
Infinite-solution rank condition
$\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf b])<n$
No-solution rank condition
$\operatorname{rank}(A)<\operatorname{rank}([A\mid\mathbf b])$
Two-line determinant
$D=a_1b_2-a_2b_1$
Nonparallel-line condition
$D\ne0\quad\Longrightarrow\quad\text{one intersection}$
Two-by-two determinant
$\det A=\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}=a_1b_2-a_2b_1$
Unique 2 by 2 system
$\det A\ne0$
Parameter case trigger
$p(k)=0\quad\text{or}\quad p(k)\ne0$
Determinant case split
$\det A(k)\ne0\Rightarrow\text{unique};\qquad \det A(k)=0\Rightarrow\text{inspect }[A\mid\mathbf b]$
Two-category total
$x+y=T$
Weighted-total model
$p_1x+p_2y=P$
Inverse-matrix solution
$\mathbf{x}=A^{-1}\mathbf{b},\qquad\det A\ne0$
Residual vector
$\mathbf r=A\widehat{\mathbf x}-\mathbf b$
Solution verification
$A\widehat{\mathbf x}=\mathbf b$
System form
$A\mathbf x=\mathbf b$
Allowed row replacement
$R_i\to R_i+\lambda R_j$
Consistency test
$\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf b])$
Unique solution test
$\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf b])=n$
Infinite solutions test
$\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf b])<n$
No solution test
$\operatorname{rank}(A)<\operatorname{rank}([A\mid\mathbf b])$
Two-by-two determinant test
$\det A=a_1b_2-a_2b_1$
Inverse method condition
$\mathbf x=A^{-1}\mathbf b,\qquad\det A\ne0$
Unit 2
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Functions
Functions are a central organizing idea in AA HL. The SL core develops function notation, transformations, inverses, quadratics and exponential/logarithmic behaviour; Higher Level extends this to general polynomials, more complicated rational functions, parity, inequalities and transformations that substantially change graph structure.
16 topics
Unit 2 progress0%
Topics in this unit
Topic 2.1Not started
Equations of straight lines
Students revisit coordinate geometry through gradient, intercepts and relationships between lines. These skills recur in tangent/normal questions in calculus.
135 min estimated
Topic 2.1 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Find a line from points, gradient or intercept information.
Recognize parallel lines from equal gradients.
Use the negative reciprocal gradient relationship for perpendicular non-vertical lines.
Key formulas
Gradient from two points
$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1},\qquad x_2\ne x_1$
Gradient-intercept form
$y=mx+c$
Intercepts of y = mx + c
$y\text{-intercept}: (0,c),\qquad x\text{-intercept}: \left(-\frac{c}{m},0\right),\ m\ne0$
Point-slope form
$y-y_1=m(x-x_1)$
Two-point equation
$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1),\qquad x_2\ne x_1$
Parallel-line condition
$m_1=m_2$
Perpendicular-line condition
$m_1m_2=-1\qquad\Longleftrightarrow\qquad m_2=-\frac{1}{m_1}$
Intercept method for $ax+by=d$
$x\text{-intercept}=\left(\frac{d}{a},0\right),\qquad y\text{-intercept}=\left(0,\frac{d}{b}\right)$
Generic linear model
$\text{output}=\text{rate}\times\text{input}+\text{initial value}$
Topic 2.2Not started
The concept of a function
A function connects permitted inputs to outputs according to a rule. Students should treat domain and range as part of the definition, not as optional graph labels.
140 min estimated
Topic 2.2 progress0%
Key ConceptsSubscription required Question BankSubscription required FlashcardsSubscription required
Learning objectives
Use notation such as f(x) and v(t).
State or interpret domain and range.
Understand inverse functions graphically and algebraically, including the reflection in $y = x$ where an inverse exists.
Key formulas
Function mapping
$f:A\to B,\qquad x\mapsto f(x)$
Output notation
$y=f(x)$
Evaluating a function
$f(a)=\left.f(x)\right|_{x=a}$
Domain notation
$\operatorname{Dom}(f)=\{x:\ f(x)\text{ is defined}\}$
Range notation
$\operatorname{Ran}(f)=\{f(x):\ x\in\operatorname{Dom}(f)\}$
Graph coordinates
$(x,y)=(x,f(x))$
Sideways parabola example
$x=y^2\quad\Longrightarrow\quad y=\pm\sqrt{x}$
Topic 2.3Not started
Graphs of functions
Students sketch and interpret graphs from equations, tables, contexts and given features. Technology may be used to explore more complicated graphs, but sketches should still communicate essential shape and behavior.
145 min estimated
Topic 2.3 progress0%
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Learning objectives
Draw a meaningful sketch from mathematical or contextual information.
Use technology to graph combinations such as sums and differences of functions.
Relate a graph back to the original variables and context.
Key formulas
Point on a function graph
$(a,b)\text{ lies on }y=f(x)\iff b=f(a)$
Domain from a graph
$\operatorname{Dom}(f)=\{x\in\mathbb{R}:\text{the graph contains a point with x-coordinate }x\}$
Range from a graph
$\operatorname{Ran}(f)=\{y\in\mathbb{R}:y=f(x)\text{ for some }x\in\operatorname{Dom}(f)\}$
x-intercepts and roots
$f(x)=0\quad\Longleftrightarrow\quad (x,0)\text{ is an x-intercept}$
y-intercept
$x=0\quad\Longrightarrow\quad (0,f(0))$
Quadratic vertex form
$f(x)=a(x-h)^2+k\quad\Longrightarrow\quad\text{vertex }(h,k)$
Sign from graph position
$f(x)>0\iff\text{graph above x-axis},\qquad f(x)<0\iff\text{graph below x-axis}$
Useful quadratic form
$y=a(x-h)^2+k$
Useful reciprocal form
$y=\frac{a}{x-h}+k$
Root as an intercept
$f(r)=0\iff (r,0)\text{ lies on }y=f(x)$
Horizontal-level solution
$f(x)=k\iff y=f(x)\text{ intersects }y=k$
Intersection condition
$f(x)=g(x)\quad\Longleftrightarrow\quad f(x)-g(x)=0$
Point on graph
$(a,b)\text{ on }y=f(x)\iff f(a)=b$
Root or x-intercept
$f(x)=0$
y-intercept
$(0,f(0))$
Intersection
$f(x)=g(x)$
Sign
$f(x)>0\text{ above the x-axis},\qquad f(x)<0\text{ below the x-axis}$
Topic 2.4Not started
Key features of graphs
A graph communicates far more than individual points. AA HL questions often require students to identify intercepts, extrema, asymptotes, intervals, symmetry or intersections and use these features to solve a problem.
150 min estimated
Topic 2.4 progress0%
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Learning objectives
Identify and interpret important graph features.
Use a GDC to locate intersections when exact algebra is not required.
Choose a graphing window that reveals the relevant behavior.
Key formulas
Point on a graph
$(a,b)\text{ lies on }y=f(x)\iff f(a)=b$
Even-root condition
$\sqrt{u(x)}\text{ is real}\iff u(x)\geq0$
Denominator condition
$\frac{p(x)}{q(x)}\text{ is defined only when }q(x)\ne0$
Logarithm condition
$\log_a(u(x))\text{ is defined only when }u(x)>0$
Root condition
$f(r)=0\iff (r,0)\text{ is an x-intercept}$
y-intercept
$(0,f(0))$
Quadratic vertex x-coordinate
$x_{\mathrm v}=-\frac{b}{2a}\quad\text{for }f(x)=ax^2+bx+c$
Positive region
$f(x)>0\iff\text{the graph is above the x-axis}$
Negative region
$f(x)<0\iff\text{the graph is below the x-axis}$
Quadratic axis of symmetry
$x=-\frac{b}{2a}$
Even symmetry
$f(-x)=f(x)$
Odd symmetry
$f(-x)=-f(x)$
Positive even-degree end behavior
$a>0,\ n\text{ even}:\quad ax^n\to\infty\text{ as }x\to\pm\infty$
Positive odd-degree end behavior
$a>0,\ n\text{ odd}:\quad ax^n\to-\infty\text{ as }x\to-\infty,\quad ax^n\to\infty\text{ as }x\to\infty$
Transformed reciprocal asymptotes
$f(x)=\frac{a}{x-h}+k\quad\Longrightarrow\quad x=h\text{ and }y=k$
Horizontal asymptote statement
$\lim_{x\to\infty}f(x)=L\quad\Longrightarrow\quad y=L\text{ is a right-end horizontal asymptote}$
Intersection condition
$f(x)=g(x)\iff f(x)-g(x)=0$
Quadratic vertex input
$x_{\mathrm v}=-\frac{b}{2a}$
Root command target
$Y_1=f(x),\qquad\text{find }x\text{ where }Y_1=0$
Intersection command target
$Y_1=f(x),\quad Y_2=g(x),\qquad\text{find where }Y_1=Y_2$
x-intercepts and roots
$f(x)=0$
Quadratic axis
$x=-\frac{b}{2a}$
Intersection
$f(x)=g(x)$
Vertical asymptote of transformed reciprocal
$f(x)=\frac{a}{x-h}+k\quad\Longrightarrow\quad x=h$
Horizontal asymptote of transformed reciprocal
$f(x)=\frac{a}{x-h}+k\quad\Longrightarrow\quad y=k$
Topic 2.5Not started
Composite and inverse functions
Composition means applying one function after another. Students should pay attention to order, because f(g(x)) and g(f(x)) are generally different.
150 min estimated
Topic 2.5 progress0%
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Learning objectives
Evaluate and simplify composite functions.
Use the identity relationship between a function and its inverse.
Check domain restrictions that may arise under composition or inversion.
Key formulas
Composite notation
$(f\circ g)(x)=f(g(x))$
Reversed order
$(g\circ f)(x)=g(f(x))$
Substitution pattern
$f(u)\text{ with }u=g(x)\quad\Longrightarrow\quad f(g(x))$
Composite domain
$\operatorname{Dom}(f\circ g)=\{x\in\operatorname{Dom}(g):g(x)\in\operatorname{Dom}(f)\}$
Reversed mapping
$f(a)=b\iff f^{-1}(b)=a$
Inverse construction
$y=f(x)\quad\xrightarrow{\text{swap }x,y}\quad x=f(y)\quad\xrightarrow{\text{solve for }y}\quad y=f^{-1}(x)$
First inverse identity
$(f\circ f^{-1})(x)=f(f^{-1}(x))=x$
Second inverse identity
$(f^{-1}\circ f)(x)=f^{-1}(f(x))=x$
Algebraic one-to-one condition
$f(a)=f(b)\Longrightarrow a=b$
Inverse domain
$\operatorname{Dom}(f^{-1})=\operatorname{Ran}(f)$
Inverse range
$\operatorname{Ran}(f^{-1})=\operatorname{Dom}(f)$
Inverse graph reflection
$(a,b)\in f\iff(b,a)\in f^{-1}$
Fixed-point condition
$f(x)=x$
Numerical inverse value
$f^{-1}(k)=a\iff f(a)=k$
Composition
$(f\circ g)(x)=f(g(x))$
Inverse mapping
$f(a)=b\iff f^{-1}(b)=a$
Inverse identities
$f(f^{-1}(x))=x,\qquad f^{-1}(f(x))=x$
Domain and range swap
$\operatorname{Dom}(f^{-1})=\operatorname{Ran}(f),\qquad\operatorname{Ran}(f^{-1})=\operatorname{Dom}(f)$
Graph reflection
$(a,b)\mapsto(b,a)\text{ across }y=x$
Topic 2.6Not started
Quadratic functions
Quadratics are studied in several equivalent forms, each revealing different graph information. Efficient students choose the form that matches the question.
165 min estimated
Topic 2.6 progress0%
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Learning objectives
Use standard form for coefficients and the y-intercept.
Use factored form for roots/x-intercepts.
Use vertex form for the turning point and transformations.
Key formulas
General quadratic form
$f(x)=ax^2+bx+c,\qquad a\ne0$
Standard form
$f(x)=ax^2+bx+c$
Factorized form
$f(x)=a(x-p)(x-q)$
Vertex form
$f(x)=a(x-h)^2+k$
y-intercept
$f(0)=c\quad\Longrightarrow\quad(0,c)$
Roots from factorized form
$a(x-p)(x-q)=0\quad\Longrightarrow\quad x=p\text{ or }x=q$
Vertex input
$x_{\mathrm v}=-\frac{b}{2a}$
Vertex
$\left(-\frac{b}{2a},\ f\left(-\frac{b}{2a}\right)\right)$
Axis from two roots
$x=\frac{p+q}{2}$
Square completion identity
$x^2+bx=\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2$
Standard-to-vertex conversion
$ax^2+bx+c=a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}$
Zero-product rule
$AB=0\quad\Longrightarrow\quad A=0\text{ or }B=0$
Quadratic formula
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Discriminant
$\Delta=b^2-4ac$
Root classification
$\Delta>0:\ 2\text{ distinct real roots};\quad\Delta=0:\ 1\text{ repeated real root};\quad\Delta<0:\ 0\text{ real roots}$
Root as graph feature
$f(x)=0\iff(x,0)\text{ lies on }y=f(x)$
Intersection equation
$f(x)=g(x)$
Upward-opening range
$a>0:\quad\operatorname{Ran}(f)=[k,\infty)$
Downward-opening range
$a<0:\quad\operatorname{Ran}(f)=(-\infty,k]$
Transformation form
$f(x)=a(x-h)^2+k$
Optimization input
$x_{\mathrm{opt}}=-\frac{b}{2a}$
Standard form
$f(x)=ax^2+bx+c,\quad a\ne0$
Axis and vertex input
$x=-\frac{b}{2a}$
Completing the square
$x^2+bx=\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2$
Topic 2.7Not started
Quadratic equations and inequalities
Students solve quadratic equations by factorization, completing the square, the quadratic formula or technology when appropriate. The discriminant provides information about the number of real roots.
175 min estimated
Topic 2.7 progress0%
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Learning objectives
Solve equations and inequalities and express the solution correctly.
Use the discriminant to classify roots.
Relate algebraic solutions to graph intersections and sign intervals.
Key formulas
Standard quadratic equation
$ax^2+bx+c=0,\quad a\ne0$
Zero-product principle
$PQ=0\quad\Longrightarrow\quad P=0\ \text{or}\ Q=0$
Roots from factored form
$a(x-p)(x-q)=0\quad\Longrightarrow\quad x=p\ \text{or}\ x=q$
Repeated factor
$a(x-p)^2=0\quad\Longrightarrow\quad x=p$
Completing-the-square identity
$x^2+bx=\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2$
Square-root step
$u^2=k,\ k>0\quad\Longrightarrow\quad u=\pm\sqrt{k}$
Quadratic formula
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Discriminant
$\Delta=b^2-4ac$
Two distinct real roots
$\Delta>0$
One repeated real root
$\Delta=0$
No real roots
$\Delta<0$
Exact-to-approximate reporting
$x=\frac{-b\pm\sqrt{\Delta}}{2a}\quad\Longrightarrow\quad x\approx x_1,\ x_2$
Graphical root condition
$f(x)=0\quad\Longleftrightarrow\quad (x,0)\ \text{lies on}\ y=f(x)$
Strict interval endpoints
$x<p\ \text{or}\ x>q\quad\Longleftrightarrow\quad (-\infty,p)\cup(q,\infty)$
Inclusive interval
$p\le x\le q\quad\Longleftrightarrow\quad [p,q]$
Union of outside intervals
$(-\infty,p)\cup(q,\infty)$
All real numbers except a point
$\mathbb{R}\setminus\{p\}=(-\infty,p)\cup(p,\infty)$
Empty solution set
$\varnothing$
Above the x-axis
$f(x)>0$
One graph above another
$f(x)>g(x)\quad\Longleftrightarrow\quad f(x)-g(x)>0$
Intersection equation
$f(x)=g(x)\quad\Longleftrightarrow\quad f(x)-g(x)=0$
Tangency condition
$\Delta=0$
GDC equation entry
$Y_1=ax^2+bx+c$
GDC intersection entry
$Y_1=f(x),\quad Y_2=g(x)$
Sum of roots check
$x_1+x_2=-\frac{b}{a}$
Product of roots check
$x_1x_2=\frac{c}{a}$
Root relationships
$x_1+x_2=-\frac{b}{a},\quad x_1x_2=\frac{c}{a}$
Interval notation reminder
$<,>:\ \text{exclude roots};\qquad \le,\ge:\ \text{include roots}$
Topic 2.8Not started
Reciprocal and rational functions
Reciprocal and simple rational functions introduce excluded values and asymptotic behavior. Students should connect the algebraic denominator with graph restrictions.
180 min estimated
Topic 2.8 progress0%
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Learning objectives
Understand the reciprocal function and its self-inverse property.
Sketch or analyze simple linear-over-linear rational functions.
Identify vertical and horizontal asymptotes from the function structure.
Key formulas
General rational function
$f(x)=\frac{P(x)}{Q(x)},\quad Q(x)\ne0$
Reciprocal identity
$x\cdot\frac{1}{x}=1,\quad x\ne0$
Basic reciprocal function
$f(x)=\frac{1}{x},\quad x\ne0$
Odd symmetry
$f(-x)=-f(x)$
Self-inverse property
$f^{-1}(x)=f(x)=\frac{1}{x}$
Parent reciprocal domain and range
$D_f=\mathbb R\setminus\{0\},\quad R_f=\mathbb R\setminus\{0\}$
Transformed reciprocal domain and range
$D_f=\mathbb R\setminus\{h\},\quad R_f=\mathbb R\setminus\{k\}$
Transformed reciprocal asymptotes
$y=\frac{a}{x-h}+k\quad\Longrightarrow\quad x=h,\ y=k$
Centre
$(h,k)$
x-intercept condition
$\frac{P(x)}{Q(x)}=0\quad\Longleftrightarrow\quad P(x)=0,\ Q(x)\ne0$
y-intercept
$(0,f(0))$
Transformed reciprocal
$f(x)=\frac{a}{x-h}+k,\quad a\ne0$
Point from an offset
$x=h+t\quad\Longrightarrow\quad y=k+\frac{a}{t},\quad t\ne0$
Linear-over-linear function
$f(x)=\frac{ax+b}{cx+d},\quad c\ne0$
Vertical asymptote candidate
$x=-\frac{d}{c}$
Horizontal asymptote
$y=\frac{a}{c}$
Hyperbola condition
$ad-bc\ne0$
Linear-over-linear transformed form
$\frac{ax+b}{cx+d}=\frac{bc-ad}{c^2\left(x+\frac{d}{c}\right)}+\frac{a}{c}$
Centre from coefficients
$\left(-\frac{d}{c},\frac{a}{c}\right)$
Cancelled-factor model
$\frac{(x-r)P(x)}{(x-r)Q(x)}=\frac{P(x)}{Q(x)},\quad x\ne r$
Rational-equation workflow
$\text{restrictions}\ \longrightarrow\ \text{clear denominators}\ \longrightarrow\ \text{solve}\ \longrightarrow\ \text{check}$
Intersection condition
$f(x)=g(x)$
Intersection coordinates
$(x,f(x))=(x,g(x))$
Inverse variation
$y=\frac{k}{x},\quad k\ne0$
Average-cost model
$C(x)=\frac{F}{x}+v$
Fixed-distance time
$t=\frac{d}{v}$
GDC entry pattern
$Y_1=\frac{P(X)}{Q(X)}$
Numerical intersection
$Y_1=Y_2$
Parent reciprocal
$f(x)=\frac1x$
Transformed reciprocal
$f(x)=\frac{a}{x-h}+k$
Transformed asymptotes
$x=h,\quad y=k$
Linear-over-linear asymptotes
$x=-\frac dc,\quad y=\frac ac$
x-intercept condition
$P(x)=0,\quad Q(x)\ne0$
Topic 2.9Not started
Exponential and logarithmic functions
Exponential and logarithmic functions are inverse families and are central to growth, decay and calculus. Students should recognize their domains, ranges and characteristic graph shapes.
190 min estimated
Topic 2.9 progress0%
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Learning objectives
Work with $a^{x}$ and $e^{x}$ .
Work with $log_{a}$ x and ln x.
Use inverse relationships to solve and interpret equations.
Key formulas
Basic exponential function
$f(x)=a^x,\qquad a>0,\ a\ne1$
Scaled exponential function
$f(x)=A a^x$
Natural exponential function
$f(x)=e^x,\qquad e\approx2.71828$
Basic exponential domain and range
$D=\mathbb R,\qquad R=(0,\infty)$
Basic exponential intercept
$a^0=1\quad\Longrightarrow\quad(0,1)$
Basic exponential asymptote
$y=0$
Discrete exponential model
$A=A_0b^t$
Growth condition
$b>1$
Decay condition
$0<b<1$
Percentage growth
$A=A_0(1+r)^n$
Percentage decay
$A=A_0(1-r)^n$
Compound interest
$A=P\left(1+\frac{r}{m}\right)^{mt}$
Continuous model
$A=A_0e^{kt}$
Discrete-continuous connection
$b=e^k,\qquad k=\ln b$
Doubling time
$T_2=\frac{\ln2}{k},\qquad k>0$
Half-life
$T_{\frac{1}{2}}=\frac{\ln2}{|k|},\qquad k<0$
Exponential-logarithmic equivalence
$\log_a x=y\quad\Longleftrightarrow\quad a^y=x$
Logarithm restrictions
$a>0,\qquad a\ne1,\qquad x>0$
Fundamental values
$\log_a1=0,\qquad\log_a a=1$
Natural inverse identities
$\ln(e^x)=x,\qquad e^{\ln x}=x\ (x>0)$
Basic logarithmic domain and range
$D=(0,\infty),\qquad R=\mathbb R$
Basic logarithmic intercept
$\log_a1=0\quad\Longrightarrow\quad(1,0)$
Basic logarithmic asymptote
$x=0$
Inverse pair
$f(x)=a^x,\qquad f^{-1}(x)=\log_a x$
Inverse compositions
$\log_a(a^x)=x,\qquad a^{\log_a x}=x\ (x>0)$
Reflected points
$(p,q)\longleftrightarrow(q,p)$
Product law
$\log_a(MN)=\log_aM+\log_aN$
Quotient law
$\log_a\left(\frac{M}{N}\right)=\log_aM-\log_aN$
Power law
$\log_a(M^p)=p\log_aM$
Argument conditions
$M>0,\qquad N>0$
Change of base
$\log_a x=\frac{\log_bx}{\log_ba}=\frac{\ln x}{\ln a}$
Common calculator form
$\log_a x=\frac{\log x}{\log a}$
Equal-base principle
$a^u=a^v\quad\Longrightarrow\quad u=v$
General exponential solution
$a^x=b\quad\Longrightarrow\quad x=\frac{\ln b}{\ln a}$
Natural exponential solution
$e^{kx}=c\quad\Longrightarrow\quad x=\frac{\ln c}{k}$
Single-log equation
$\log_a F(x)=c\quad\Longleftrightarrow\quad F(x)=a^c$
Equal-log principle
$\log_aM=\log_aN\quad\Longrightarrow\quad M=N$
Logarithmic domain
$M>0,\qquad N>0$
Transformed exponential
$y=Aa^{B(x-h)}+k$
Exponential asymptote
$y=k$
Transformed logarithm
$y=A\log_a(B(x-h))+k$
Logarithmic domain condition
$B(x-h)>0$
Time to a target
$A_0b^t=T\quad\Longrightarrow\quad t=\frac{\ln\left(\frac{T}{A_0}\right)}{\ln b}$
Continuous time to a target
$A_0e^{kt}=T\quad\Longrightarrow\quad t=\frac{\ln\left(\frac{T}{A_0}\right)}{k}$
Half-life model
$A=A_0\left(\frac{1}{2}\right)^{\frac{t}{h}}$
Intersection method
$Y_1=F(x),\qquad Y_2=G(x)$
Root method
$Y_1=F(x)-G(x)$
Approximation notation
$x=\frac{\ln7}{\ln3}-1\approx0.771$
Logarithmic validity check
$\text{every original argument}>0$
Topic 2.10Not started
Solving equations graphically and analytically
AA HL encourages students to choose the most efficient representation. Some equations are best solved symbolically; others are naturally solved by finding graph intersections or roots using technology.
200 min estimated
Topic 2.10 progress0%
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Learning objectives
Solve equations by algebra when an exact method is accessible.
Use graphs or numerical methods appropriately when technology is expected.
Check the solution against domain and context.
Key formulas
Equation solution condition
$L(x)=R(x)$
Root form
$f(x)=0$
Equivalent intersection form
$F(x)=G(x)\quad\Longleftrightarrow\quad F(x)-G(x)=0$
Intersection equation
$Y_1=F(x),\qquad Y_2=G(x)$
Equivalent zero equation
$Y_1=F(x)-G(x)$
Root coordinate
$f(r)=0\quad\Longleftrightarrow\quad(r,0)\text{ lies on }y=f(x)$
Addition property
$A=B\quad\Longrightarrow\quad A+C=B+C$
Non-zero multiplication property
$A=B\quad\Longrightarrow\quad kA=kB,\qquad k\ne0$
Verification
$L(x_{\mathrm{candidate}})\stackrel{?}{=}R(x_{\mathrm{candidate}})$
General linear equation
$ax+b=cx+d$
Linear solution
$x=\frac{d-b}{a-c},\qquad a\ne c$
Parallel-line condition
$a=c,\qquad b\ne d$
Quadratic equation
$ax^2+bx+c=0,\qquad a\ne0$
Quadratic formula
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Discriminant
$\Delta=b^2-4ac$
Root classification
$\Delta>0:\ 2\text{ roots};\quad\Delta=0:\ 1\text{ repeated root};\quad\Delta<0:\ 0\text{ real roots}$
Root condition
$f(r)=0$
Crossing sign test
$f(a)f(b)<0$
Repeated quadratic factor
$f(x)=(x-r)^2g(x)$
Intersection condition
$F(x)=G(x)$
Intersection point
$\bigl(r,F(r)\bigr)=\bigl(r,G(r)\bigr)$
Restricted solution set
$\{x:F(x)=G(x)\}\cap D$
Equal-base principle
$a^u=a^v\quad\Longrightarrow\quad u=v$
Logarithmic solution
$a^x=b\quad\Longrightarrow\quad x=\frac{\ln b}{\ln a}$
Positive substitution
$u=a^x,\qquad u>0$
Logarithmic equivalence
$\log_aF(x)=c\quad\Longleftrightarrow\quad F(x)=a^c$
Equal-log principle
$\log_aM=\log_aN\quad\Longrightarrow\quad M=N$
Domain requirement
$M>0,\qquad N>0$
Rational restriction
$D(x)\ne0$
Zero of a rational expression
$\frac{N(x)}{D(x)}=0\quad\Longleftrightarrow\quad N(x)=0\text{ and }D(x)\ne0$
Cross multiplication condition
$\frac{A}{B}=\frac{C}{D}\quad\Longrightarrow\quad AD=BC,\qquad B\ne0,\ D\ne0$
Exact-to-approximate notation
$x=\frac{-1+\sqrt{33}}{4}\approx1.19$
Absolute checking error
$\left|L(x_{\mathrm{approx}})-R(x_{\mathrm{approx}})\right|$
Factor route
$P(x)=0\quad\longrightarrow\quad\text{factor if practical}$
Intersection route
$F(x)=G(x)\quad\longrightarrow\quad\text{graph }Y_1=F(x),\ Y_2=G(x)$
Logarithm route
$a^{g(x)}=b\quad\longrightarrow\quad g(x)=\frac{\ln b}{\ln a}$
Residual
$R(x)=F(x)-G(x)$
Exact check
$R(r)=0$
Approximate check
$|R(r)|\approx0$
One-graph zero setup
$Y_1=F(x)-G(x)$
Two-graph intersection setup
$Y_1=F(x),\qquad Y_2=G(x)$
Bracket condition
$f(a)f(b)<0$
Method-check structure
$\text{classify}\to\text{restrict}\to\text{solve}\to\text{check}\to\text{report}$
Approximation notation
$x\approx r\quad\text{when }r\text{ is rounded}$
Topic 2.11Not started
Transformations of graphs
Transformations allow students to predict a new graph from a known one. The most common errors come from confusing horizontal and vertical effects or applying transformations in the wrong order.
200 min estimated
Topic 2.11 progress0%
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Learning objectives
Translate graphs horizontally and vertically.
Reflect in the coordinate axes.
Apply vertical and horizontal stretches/compressions and combine transformations.
Key formulas
Parent point condition
$v=f(u)$
General transformed function
$g(x)=a f\bigl(b(x-h)\bigr)+k$
General point mapping
$(u,v)\longmapsto\left(h+\frac{u}{b},\ k+av\right),\qquad b\ne0$
Vertical translation
$g(x)=f(x)+k$
Vertical point map
$(u,v)\longmapsto(u,v+k)$
Range translation
$R_g=\{v+k:v\in R_f\}$
Horizontal translation
$g(x)=f(x-h)$
Horizontal point map
$(u,v)\longmapsto(u+h,v)$
Domain translation
$D_g=\{u+h:u\in D_f\}$
Reflection in the $x$-axis
$g(x)=-f(x)$
$x$-axis point map
$(u,v)\longmapsto(u,-v)$
Reflection in the $y$-axis
$g(x)=f(-x)$
$y$-axis point map
$(u,v)\longmapsto(-u,v)$
Vertical scaling
$g(x)=af(x)$
Vertical scale point map
$(u,v)\longmapsto(u,av)$
Vertical stretch condition
$|a|>1$
Vertical compression condition
$0<|a|<1$
Horizontal scaling
$g(x)=f(bx),\qquad b\ne0$
Horizontal scale point map
$(u,v)\longmapsto\left(\frac{u}{b},v\right)$
Horizontal scale factor
$\frac{1}{|b|}$
Combined form
$g(x)=a f\bigl(b(x-h)\bigr)+k$
Combined point map
$(u,v)\longmapsto\left(h+\frac{u}{b},\ k+av\right)$
Inverse point recovery
$u=b(x-h),\qquad v=\frac{y-k}{a}$
Visual transformation
$g(x)=-2f(x-3)+1$
Visual point map
$(u,v)\longmapsto(u+3,1-2v)$
Transformed quadratic range
$R_g=(-\infty,1]$
Transformed domain
$D_g=\left\{h+\frac{u}{b}:u\in D_f\right\}$
Transformed range
$R_g=\{k+av:v\in R_f\}$
Horizontal asymptote mapping
$y=L\quad\longmapsto\quad y=k+aL$
Vertical asymptote mapping
$x=C\quad\longmapsto\quad x=h+\frac{C}{b}$
Transformed $y$-intercept
$g(0)=a f(-bh)+k$
Transformed zeros
$a f\bigl(b(x-h)\bigr)+k=0$
Inside factorization
$bx+c=b\left(x+\frac{c}{b}\right)$
Readable inside form
$b(x-h)$
Quadratic transformed form
$g(x)=a(x-h)^2+k$
Absolute-value transformed form
$g(x)=a|x-h|+k$
Reciprocal translated form
$g(x)=\frac{a}{x-h}+k$
Sketch workflow
$\text{parent}\to\text{asymptotes or boundaries}\to\text{landmarks}\to\text{intercepts}\to\text{shape}$
GDC parent entry
$Y_1=f(x)$
GDC image entry
$Y_2=a f\bigl(b(x-h)\bigr)+k$
Parameter checklist
$a:\text{ vertical};\quad b:\text{ reciprocal horizontal};\quad h:\text{ horizontal shift};\quad k:\text{ vertical shift}$
Reliable point check
$g\left(h+\frac{u}{b}\right)=k+a f(u)$
Topic 2.12Not started
Polynomial functions, roots and the factor/remainder theorems
Higher Level treats polynomial structure more generally. Students move between factors, zeros, coefficients and graph behaviour and use the factor and remainder theorems to avoid unnecessary long division.
80 min estimated
Topic 2.12 progress0%
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Learning objectives
Identify the degree, leading coefficient, roots, multiplicities and end behaviour of a polynomial function.
Move fluently between roots, linear factors, intercepts and a polynomial equation.
Apply the factor theorem and remainder theorem, including divisors of the form ax-b.
Divide polynomials using long division or synthetic division and verify the quotient-remainder identity.
Use factor and remainder conditions to determine unknown coefficients.
Factorize and solve higher-degree polynomial equations exactly where possible.
Use relationships between roots and coefficients to solve and construct polynomial problems.
Sketch polynomial graphs and use a GDC to find, check and interpret numerical roots.
Key formulas
General polynomial
$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\qquad a_n\ne0$
Leading-term approximation
$f(x)\sim a_nx^n\quad\text{as }|x|\to\infty$
Root-factor connection
$f(r)=0\iff(x-r)\text{ is a factor of }f(x)$
Factored polynomial
$f(x)=a\prod_{k=1}^{n}(x-r_k)$
Multiplicity definition
$(x-r)^m\mid f(x),\qquad (x-r)^{m+1}\nmid f(x)$
Factor theorem
$(x-a)\mid f(x)\iff f(a)=0$
General linear factor
$(ax-b)\mid f(x)\iff f\!\left(\frac{b}{a}\right)=0$
Remainder theorem
$f(x)=(x-a)q(x)+f(a)$
Remainder for ax-b
$R=f\!\left(\frac{b}{a}\right)$
Division algorithm
$f(x)=d(x)q(x)+r(x),\qquad \deg r<\deg d$
Factor condition
$(x-a)\mid f(x)\Longrightarrow f(a)=0$
Remainder condition
$f(x)\div(x-a)\text{ leaves }R\Longrightarrow f(a)=R$
Repeated-root conditions
$(x-a)^2\mid f(x)\Longrightarrow f(a)=0\text{ and }f'(a)=0$
Rational-root candidates
$r=\frac{p}{q},\qquad p\mid a_0,\quad q\mid a_n$
Conjugate factor pair
$(x-(a+bi))(x-(a-bi))=(x-a)^2+b^2$
General root sum
$\sum\limits_{k=1}^{n}r_k=-\frac{a_{n-1}}{a_n}$
General root product
$\prod_{k=1}^{n}r_k=(-1)^n\frac{a_0}{a_n}$
Cubic relationships
$\alpha+\beta+\gamma=-\frac{b}{a},\quad \alpha\beta+\alpha\gamma+\beta\gamma=\frac{c}{a},\quad \alpha\beta\gamma=-\frac{d}{a}$
Maximum number of turning points
$N_{\mathrm{turn}}\le n-1$
y-intercept
$(0,f(0))$
Quadratic formula
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Positive even degree
$a_n>0,\ n\text{ even}:\quad f(x)\to+\infty\text{ as }x\to\pm\infty$
Positive odd degree
$a_n>0,\ n\text{ odd}:\quad f(x)\to-\infty\text{ as }x\to-\infty,\ f(x)\to+\infty\text{ as }x\to+\infty$
Remainder theorem
$R=f(a)\quad\text{for division by }x-a$
Division algorithm
$f=dq+r,\qquad \deg r<\deg d$
Root sum and product
$\sum\limits_{k=1}^{n}r_k=-\frac{a_{n-1}}{a_n},\qquad \prod_{k=1}^{n}r_k=(-1)^n\frac{a_0}{a_n}$
Maximum turning points
$N_{\mathrm{turn}}\le n-1$
Topic 2.13Not started
Higher-degree rational functions and asymptotes
Rational functions at HL can have more complicated numerator and denominator degrees, leading to several vertical asymptotes and to horizontal or oblique asymptotic behaviour. Algebraic form and graph features must be interpreted together.
85 min estimated
Topic 2.13 progress0%
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Learning objectives
Analyze domains, zeros, intercepts and asymptotes of rational functions.
Use factorization to identify denominator zeros and possible cancellations.
Use degree comparison or polynomial division to determine end behaviour and oblique asymptotes.
Construct or interpret a graph using algebraic information rather than relying only on technology.
Solve equations and inequalities involving rational functions while respecting excluded values.
Key formulas
General rational function
$f(x)=\frac{P(x)}{Q(x)},\qquad Q(x)\ne0$
Degree notation
$m=\deg P,\qquad n=\deg Q$
Domain rule
$\operatorname{Dom}(f)=\{x\in\mathbb{R}:Q(x)\ne0\}$
x-intercept condition
$P(r)=0\quad\text{and}\quad Q(r)\ne0$
y-intercept
$\left(0,\frac{P(0)}{Q(0)}\right),\qquad Q(0)\ne0$
Vertical-asymptote test
$Q_s(a)=0,\ P_s(a)\ne0\quad\Longrightarrow\quad x=a$
One-sided notation
$\lim_{x\to a^-}f(x),\qquad \lim_{x\to a^+}f(x)$
Hole coordinate
$x=a\text{ cancelled},\qquad \text{hole}=\bigl(a,f_s(a)\bigr)$
Lower numerator degree
$\deg P<\deg Q\quad\Longrightarrow\quad y=0$
Equal degrees
$\deg P=\deg Q\quad\Longrightarrow\quad y=\frac{\operatorname{lc}(P)}{\operatorname{lc}(Q)}$
Division identity
$P(x)=Q(x)D(x)+R(x),\qquad \deg R<\deg Q$
Asymptotic form
$f(x)=D(x)+\frac{R(x)}{Q(x)},\qquad \frac{R(x)}{Q(x)}\to0$
Polynomial asymptote criterion
$\lim_{x\to\pm\infty}\bigl(f(x)-D(x)\bigr)=0$
Sketching checkpoints
$\text{domain}\to\text{holes}\to\text{intercepts}\to\text{asymptotes}\to\text{signs}$
Cross-multiplication with restrictions
$\frac{A}{B}=\frac{C}{D}\Longrightarrow AD=BC,\qquad B\ne0,\ D\ne0$
Critical values
$P(x)=0\quad\text{or}\quad Q(x)=0$
Intersection equation
$\frac{P(x)}{Q(x)}=g(x),\qquad Q(x)\ne0$
Repeated quadratic root
$b^2-4ac=0$
Numerical asymptote check
$f(x)-D(x)\to0\quad\text{as}\quad x\to\pm\infty$
Domain
$Q(x)\ne0$
Horizontal asymptote
$m<n:y=0;\qquad m=n:y=\frac{a_m}{b_n}$
Polynomial asymptote
$f(x)=D(x)+\frac{R(x)}{Q(x)},\qquad \deg R<\deg Q$
Vertical asymptote
$Q_s(a)=0,\quad P_s(a)\ne0\quad\Longrightarrow\quad x=a$
Topic 2.14Not started
Odd/even functions and inverse-function restrictions
Symmetry becomes an algebraic property at HL. Students identify even and odd functions from equations and graphs and determine when a function must be restricted before an inverse can exist.
80 min estimated
Topic 2.14 progress0%
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Learning objectives
Test whether a function is even, odd or neither using f(-x).
Connect even functions with y-axis symmetry and odd functions with rotational symmetry about the origin.
Find inverse functions algebraically and state the corresponding domain and range.
Restrict a non-one-to-one function to a suitable domain before constructing its inverse.
Recognize and work with self-inverse functions.
Key formulas
Even function
$f(-x)=f(x)$
Domain symmetry requirement
$x\in\operatorname{Dom}(f)\iff -x\in\operatorname{Dom}(f)$
Odd function
$f(-x)=-f(x)$
Origin condition
$0\in\operatorname{Dom}(f)\text{ and }f\text{ odd}\Longrightarrow f(0)=0$
Parity decision
$f(-x)=f(x):\text{ even};\quad f(-x)=-f(x):\text{ odd}$
Even point symmetry
$(x,y)\mapsto(-x,y)$
Odd point symmetry
$(x,y)\mapsto(-x,-y)$
Inverse point reflection
$(x,y)\mapsto(y,x)$
Even part
$f_e(x)=\frac{f(x)+f(-x)}{2}$
Odd part
$f_o(x)=\frac{f(x)-f(-x)}{2}$
Decomposition
$f(x)=f_e(x)+f_o(x)$
Injectivity test
$f(a)=f(b)\Longrightarrow a=b$
Inverse existence
$f^{-1}\text{ is a function}\iff f\text{ is one-to-one}$
Right-branch inverse
$f(x)=x^2,\ x\ge0\quad\Longrightarrow\quad f^{-1}(x)=\sqrt{x}$
Left-branch inverse
$f(x)=x^2,\ x\le0\quad\Longrightarrow\quad f^{-1}(x)=-\sqrt{x}$
Domain-range swap
$\operatorname{Dom}(f^{-1})=\operatorname{Ran}(f),\qquad \operatorname{Ran}(f^{-1})=\operatorname{Dom}(f)$
Inverse workflow
$y=f(x)\to x=f(y)\to y=f^{-1}(x)$
Composition checks
$f^{-1}(f(x))=x,\qquad f(f^{-1}(x))=x$
Reflection rule
$(a,b)\in f\quad\Longleftrightarrow\quad(b,a)\in f^{-1}$
Mutual inverse check
$f\circ f^{-1}=I,\qquad f^{-1}\circ f=I$
Self-inverse condition
$f(f(x))=x$
Graph check
$y=f(x),\qquad y=f^{-1}(x),\qquad y=x$
Even
$f(-x)=f(x)$
Odd
$f(-x)=-f(x)$
One-to-one
$f(a)=f(b)\Longrightarrow a=b$
Domain-range swap
$\operatorname{Dom}(f^{-1})=\operatorname{Ran}(f),\quad \operatorname{Ran}(f^{-1})=\operatorname{Dom}(f)$
Inverse check
$f^{-1}\circ f=I,\qquad f\circ f^{-1}=I$
Topic 2.15Not started
Inequalities involving functions
Function inequalities require students to find intervals rather than isolated roots. Graphical and analytic approaches should agree and excluded points must be treated carefully.
95 min estimated
Topic 2.15 progress0%
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Learning objectives
Translate function inequalities into sign, interval and graph questions.
Solve polynomial and rational inequalities with correct critical values, endpoint decisions and domain restrictions.
Compare two functions by analyzing the sign of their difference and confirm solutions graphically.
Handle absolute-value, exponential and logarithmic inequalities relevant to AA HL.
Use a GDC effectively while presenting sufficient analytic reasoning and checking the final solution set.
Key formulas
Basic sign forms
$f(x)\gt 0,\quad f(x)\lt 0,\quad f(x)\ge0,\quad f(x)\le0$
Comparison reduction
$f(x)\ \square\ g(x)\iff f(x)-g(x)\ \square\ 0$
Critical-value set
$\{x:N(x)=0\}\cup\{x:D(x)=0\}$
Interval constancy
$\text{No zero or discontinuity in }I\Longrightarrow \operatorname{sgn}f(x)\text{ is constant on }I$
Separated sets
$I_1\cup I_2$
Simultaneous conditions
$I_1\cap I_2$
Factorized polynomial
$P(x)=a\prod_{k=1}^{n}(x-r_k)^{m_k}$
Sign-change rule
$m\text{ odd}\Rightarrow\text{sign changes};\qquad m\text{ even}\Rightarrow\text{sign is unchanged}$
Rational sign form
$\frac{N(x)}{D(x)}\ \square\ 0,\qquad D(x)\ne0$
Common real-domain conditions
$D(x)\ne0,\qquad \sqrt{u(x)}:\ u(x)\ge0,\qquad \log_a u(x):\ u(x)\gt 0$
Final restriction
$S_{\text{final}}=S_{\text{inequality}}\cap\operatorname{Dom}(f)$
Difference method
$f(x)\ \square\ g(x)\iff h(x)=f(x)-g(x)\ \square\ 0$
Graph reading
$f(x)\le g(x)\Longleftrightarrow y=f(x)\text{ is on or below }y=g(x)$
Inside rule
$|u|\le k\iff -k\le u\le k\quad(k\ge0)$
Outside rule
$|u|\ge k\iff u\le-k\text{ or }u\ge k\quad(k\ge0)$
Increasing base
$a\gt 1:\quad a^u\ \square\ a^v\iff u\ \square\ v$
Decreasing base
$0\lt a\lt 1:\quad a^u\ \square\ a^v\iff u\ \text{has the reversed comparison to }v$
Difference graph
$Y_3=Y_1-Y_2$
IB presentation chain
$\text{domain}\to\text{standard form}\to\text{critical values}\to\text{signs}\to\text{interval answer}$
Final answer check
$\text{test one interior point from every reported interval in the original inequality}$
Comparison
$f\ \square\ g\iff f-g\ \square 0$
Polynomial signs
$P(x)=a\prod(x-r_i)^{m_i}$
Rational domain
$\frac{N(x)}{D(x)}:\ D(x)\ne0$
Absolute value inside
$|u|\le k\iff-k\le u\le k$
Absolute value outside
$|u|\ge k\iff u\le-k\text{ or }u\ge k$
Topic 2.16Not started
Special graph constructions and transformations
HL transformations include operations that can fold, reflect, reciprocate or square parts of an existing graph. Students should predict new zeros, asymptotes, symmetry and range from the original graph without always starting again from an equation.
100 min estimated
Topic 2.16 progress0%
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Learning objectives
Describe translations, reflections and stretches using equations and point mappings.
Construct absolute-output, absolute-input, squared-output and reciprocal graphs from a known graph.
Combine transformations in a controlled order and identify them from a transformed equation.
Predict how transformations affect roots, intercepts, domain, range, symmetry and asymptotes.
Build and interpret piecewise or composite graph constructions relevant to AA HL.
Use a GDC to confirm an accurate hand sketch while communicating a complete IB HL method.
Key formulas
General affine transformation
$g(x)=A f(B(x-h))+k$
Point map
$(u,v)\mapsto\left(h+\frac{u}{B},\ k+Av\right)$
Vertical shift
$y=f(x)+k:\quad (u,v)\mapsto(u,v+k)$
Horizontal shift
$y=f(x-h):\quad (u,v)\mapsto(u+h,v)$
Reflection in x-axis
$y=-f(x):\quad(u,v)\mapsto(u,-v)$
Reflection in y-axis
$y=f(-x):\quad(u,v)\mapsto(-u,v)$
Vertical scaling
$y=A f(x):\quad(u,v)\mapsto(u,Av)$
Horizontal scaling
$y=f(Bx):\quad(u,v)\mapsto\left(\frac{u}{B},v\right)$
Combined map
$g(x)=A f(B(x-h))+k:\quad(u,v)\mapsto\left(h+\frac uB,k+Av\right)$
Input normalization
$Bx+C=B\left(x+\frac CB\right)$
Absolute-output definition
$|f(x)|=\begin{cases}f(x),&f(x)\ge0,\\-f(x),&f(x)\lt0.\end{cases}$
Range restriction
$|f(x)|\ge0$
Absolute-input definition
$f(|x|)=\begin{cases}f(x),&x\ge0,\\f(-x),&x\lt0.\end{cases}$
Symmetry
$f(|-x|)=f(|x|)$
Squared output
$g(x)=[f(x)]^2$
Point map
$(u,v)\mapsto(u,v^2)$
Reciprocal graph
$g(x)=\frac1{f(x)}$
Reciprocal domain
$\operatorname{Dom}(g)=\{x\in\operatorname{Dom}(f):f(x)\ne0\}$
Affine asymptote map
$x=a\mapsto x=h+\frac aB,\qquad y=b\mapsto y=k+Ab$
Reciprocal zero rule
$f(r)=0\Longrightarrow x=r\text{ is excluded from }\frac1f$
Maximum construction
$\max(f,g)=\frac{f+g+|f-g|}{2}$
Minimum construction
$\min(f,g)=\frac{f+g-|f-g|}{2}$
Construction checklist
$\text{anchors}\to\text{map}\to\text{branches}\to\text{domain/range check}$
Absolute transformed output
$(u,v)\mapsto\left(h+\frac uB,\ |Av+k|\right)$
Reciprocal level rule
$f(x)=c\Longrightarrow \frac1{f(x)-c}\text{ is undefined}$
Useful GDC levels
$f(x)=0,\qquad f(x)=1,\qquad f(x)=-1$
IB sketch workflow
$\text{formula}\to\text{domain}\to\text{map}\to\text{anchors}\to\text{branches}\to\text{check}$
Practice validation
$\text{source features}\to\text{construction rule}\to\text{new features}\to\text{final sketch}$
Affine map
$(u,v)\mapsto\left(h+\frac uB,k+Av\right)$
Absolute output
$(u,v)\mapsto(u,|v|)$
Absolute input
$f(|x|)=f(x)\text{ for }x\ge0,\ f(-x)\text{ for }x\lt0$
Squared output
$(u,v)\mapsto(u,v^2)$
Reciprocal output
$(u,v)\mapsto\left(u,\frac1v\right),\quad v\ne0$
Unit 3
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Geometry and Trigonometry
Geometry and Trigonometry is the largest HL topic by recommended teaching time. The SL core develops triangle trigonometry, radians, exact values, identities and circular functions; the HL extension adds reciprocal and inverse trigonometric functions, compound-angle identities and an extensive three-dimensional vector treatment of lines, planes, intersections and angles.
18 topics
Unit 3 progress0%
Topics in this unit
Topic 3.1Not started
Three-dimensional geometry
AA HL extends coordinate and mensuration skills into three dimensions. Students should be comfortable moving between diagrams, coordinates and formulas for distance, midpoint, area and volume.
190 min estimated
Topic 3.1 progress0%
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Learning objectives
Find the distance and midpoint between points in 3D.
Calculate surface area and volume for common solids and composite solids.
Interpret angles between intersecting lines or between a line and a plane in geometric situations.
Key formulas
Euler relation for a convex polyhedron
$V-E+F=2$
Perpendicular line-plane angle
$\theta=90^\circ$
Line-plane angle definition
$\theta=\angle(\text{line},\text{ perpendicular projection on plane})$
Pythagorean theorem
$c^2=a^2+b^2$
Rectangular base diagonal
$p=\sqrt{l^2+w^2}$
Cuboid space diagonal
$d=\sqrt{l^2+w^2+h^2}$
Distance in three dimensions
$AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
Midpoint in three dimensions
$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$
Distance from the origin
$OP=\sqrt{x^2+y^2+z^2}$
Line-plane angle using tangent
$\tan\theta=\frac{\text{perpendicular height}}{\text{projection length}}$
Line-plane angle using sine
$\sin\theta=\frac{\text{perpendicular height}}{\text{line length}}$
Line-plane angle using cosine
$\cos\theta=\frac{\text{projection length}}{\text{line length}}$
Cosine rule for an included angle
$\cos C=\frac{a^2+b^2-c^2}{2ab}$
Square centre-to-corner distance
$p=\frac{s\sqrt2}{2}$
Cuboid surface area
$SA=2(lw+lh+wh)$
Closed cylinder surface area
$SA=2\pi r^2+2\pi rh$
Sphere surface area
$SA=4\pi r^2$
Closed cone surface area
$SA=\pi r^2+\pi rs$
Prism volume
$V=A_{\text{cross-section}}L$
Cuboid volume
$V=lwh$
Cylinder volume
$V=\pi r^2h$
Pyramid volume
$V=\frac13Ah$
Cone volume
$V=\frac13\pi r^2h$
Sphere volume
$V=\frac43\pi r^3$
Diagram-to-method chain
$\text{solid}\to\text{projection or cross-section}\to\text{2D triangle}\to\text{formula}$
Cubic metres to litres
$1\text{ m}^3=1000\text{ L}$
Maximum straight length in a cuboid
$d_{\max}=\sqrt{l^2+w^2+h^2}$
Inverse tangent entry
$\theta=\tan^{-1}\left(\frac{h}{p}\right)$
Inverse sine entry
$\theta=\sin^{-1}\left(\frac{h}{d}\right)$
Reliable 3D workflow
$\text{label}\to\text{project or cross-section}\to\text{calculate}\to\text{check}\to\text{interpret}$
Topic 3.2Not started
Trigonometry in triangles
Students use right-triangle ratios as well as the sine and cosine rules for non-right triangles. Choosing the correct method is as important as substituting accurately.
205 min estimated
Topic 3.2 progress0%
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Learning objectives
Use sine, cosine and tangent in right triangles.
Use the sine rule when a matching side-angle pair is available.
Use the cosine rule for SAS/SSS-style information and use the area formula for two sides and the included angle.
Key formulas
Interior angle sum
$A+B+C=180^\circ$
Standard opposite-side convention
$a\leftrightarrow A,\quad b\leftrightarrow B,\quad c\leftrightarrow C$
Sine ratio
$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
Cosine ratio
$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
Tangent ratio
$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$
Pythagorean theorem
$a^2+b^2=c^2$
Missing shorter side
$a=\sqrt{c^2-b^2}$
Complementary acute angles
$\alpha+\beta=90^\circ$
Sine rule
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Reciprocal sine-rule form
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
Supplementary sine solution
$B_2=180^\circ-B_1$
SSA comparison height
$h=b\sin A$
Cosine rule for a side
$c^2=a^2+b^2-2ab\cos C$
Cosine rule for an angle
$\cos C=\frac{a^2+b^2-c^2}{2ab}$
Triangle area from two sides and included angle
$K=\frac{1}{2}ab\sin C$
Equivalent area forms
$K=\frac{1}{2}bc\sin A=\frac{1}{2}ca\sin B$
Method-selection guide
$\begin{aligned}\text{right triangle}&\Rightarrow\text{SOH-CAH-TOA or Pythagoras}\\\text{known opposite pair}&\Rightarrow\text{sine rule}\\\text{SAS or SSS}&\Rightarrow\text{cosine rule}\end{aligned}$
Triangle inequality
$a+b>c,\quad b+c>a,\quad c+a>b$
Right-triangle ratios
$\sin\theta=\frac{O}{H},\quad\cos\theta=\frac{A}{H},\quad\tan\theta=\frac{O}{A}$
Cosine rule
$c^2=a^2+b^2-2ab\cos C$
Triangle area
$K=\frac12 ab\sin C$
Topic 3.3Not started
Applications of trigonometry
Applied trigonometry requires students to convert words into a reliable diagram before calculating. Many errors occur because a correct formula is applied to an incorrectly labelled picture.
215 min estimated
Topic 3.3 progress0%
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Learning objectives
Use Pythagoras in two- and three-dimensional settings.
Solve elevation, depression and bearing-style geometry as appropriate.
Construct and label diagrams before selecting a trigonometric method.
Key formulas
Triangle angle sum
$A+B+C=180^\circ$
Height from elevation angle
$h=d\tan\theta$
Horizontal distance from depression angle
$d=\frac{h}{\tan\theta}$
Two-position height relation
$x\tan\beta=(x+s)\tan\alpha$
Reverse bearing
$\theta_{\text{reverse}}=(\theta+180^\circ)\bmod 360^\circ$
Bearing from east and north components
$\theta=\tan^{-1}\left(\frac{E}{N}\right)$
Cosine rule for a route displacement
$d^2=p^2+q^2-2pq\cos C$
Sine rule for triangulation
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Sine rule
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Second possible sine-rule angle
$B_2=180^\circ-B_1$
Cosine rule for a side
$c^2=a^2+b^2-2ab\cos C$
Cosine rule for an angle
$C=\cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right)$
Triangle area
$K=\frac{1}{2}ab\sin C$
Equivalent area forms
$K=\frac12 bc\sin A=\frac12 ca\sin B$
Cuboid space diagonal
$d=\sqrt{l^2+w^2+h^2}$
Line-plane angle
$\tan\theta=\frac{\text{perpendicular height}}{\text{projection length}}$
Three-dimensional distance
$d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}$
Route displacement bounds
$|p-q|\le d\le p+q$
Right-triangle ratios
$\sin\theta=\frac{O}{H},\quad\cos\theta=\frac{A}{H},\quad\tan\theta=\frac{O}{A}$
Pythagorean theorem
$a^2+b^2=c^2$
Cosine rule
$c^2=a^2+b^2-2ab\cos C$
Triangle area
$K=\frac12 ab\sin C$
Reverse bearing
$\theta_{\text{reverse}}=(\theta+180^\circ)\bmod360^\circ$
Topic 3.4Not started
Radian measure, arcs and sectors
Radians become the natural angle unit for higher mathematics and calculus. Students should switch comfortably between degree and radian thinking and use radian-based arc and sector formulas.
210 min estimated
Topic 3.4 progress0%
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Learning objectives
Represent angles in radians.
Find arc length from radius and central angle.
Find sector area from radius and central angle.
Key formulas
Radian definition
$\theta=\frac{s}{r}$
Full turn
$360^\circ=2\pi\text{ rad}$
Half turn
$180^\circ=\pi\text{ rad}$
Coterminal angles
$\theta_{\text{coterminal}}=\theta+2k\pi,\quad k\in\mathbb{Z}$
Degrees to radians
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180}$
Radians to degrees
$\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
One-radian approximation
$1\text{ rad}=\frac{180^\circ}{\pi}\approx57.3^\circ$
Standard exact angles
$30^\circ=\frac{\pi}{6},\quad45^\circ=\frac{\pi}{4},\quad60^\circ=\frac{\pi}{3},\quad90^\circ=\frac{\pi}{2}$
Arc length in radians
$s=r\theta$
Arc length in degrees
$s=\frac{\theta^\circ}{360^\circ}(2\pi r)$
Major central angle
$\theta_{\text{major}}=2\pi-\theta_{\text{minor}}$
Sector area in radians
$K=\frac12r^2\theta$
Sector area in degrees
$K=\frac{\theta^\circ}{360^\circ}\pi r^2$
Major sector area
$K_{\text{major}}=\pi r^2-K_{\text{minor}}$
Radius from an arc
$r=\frac{s}{\theta}$
Angle from an arc
$\theta=\frac{s}{r}$
Radius from sector area
$r=\sqrt{\frac{2K}{\theta}}$
Angle from sector area
$\theta=\frac{2K}{r^2}$
Sector area from radius and arc
$K=\frac12rs$
Sector perimeter
$P=2r+r\theta$
Chord length
$c=2r\sin\left(\frac{\theta}{2}\right)$
Central triangle area
$K_{\triangle}=\frac12r^2\sin\theta$
Minor segment area
$K_{\text{segment}}=\frac12r^2(\theta-\sin\theta)$
Annular sector area
$K_{\text{annular}}=\frac12(R^2-r^2)\theta$
Annular-sector perimeter
$P=(R+r)\theta+2(R-r)$
Angular displacement after turns
$\theta=2\pi n$
Rolling distance
$s=r\theta=2\pi rn$
Angle conversions
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180},\quad\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
Arc length
$s=r\theta$
Sector area
$K=\frac12r^2\theta=\frac12rs$
Topic 3.5Not started
Unit-circle definitions and exact values
The unit circle extends trigonometric ratios beyond acute angles and makes signs and exact values systematic. It also supports equation solving and graph interpretation.
205 min estimated
Topic 3.5 progress0%
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Learning objectives
Define cosine and sine as coordinates on the unit circle.
Use $\tan θ = \frac{\sin θ}{\cos θ}$ where defined.
Know exact values for standard angles and use symmetry/quadrants to extend them.
Recognize the ambiguous case when the sine rule can produce two possible triangles.
Key formulas
Unit-circle equation
$x^2+y^2=1$
Coterminal angles
$\theta_{\mathrm{coterminal}}=\theta+2k\pi,\quad k\in\mathbb{Z}$
Unit-circle coordinates
$P=(\cos\theta,\sin\theta)$
Pythagorean identity
$\sin^2\theta+\cos^2\theta=1$
Coordinate definitions
$\cos\theta=x,\quad\sin\theta=y$
Tangent identity
$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{y}{x},\quad\cos\theta\ne0$
Undefined tangent angles
$\theta=\frac{\pi}{2}+k\pi,\quad k\in\mathbb{Z}$
Quadrant I
$\sin\theta>0,\quad\cos\theta>0,\quad\tan\theta>0$
Quadrant II
$\sin\theta>0,\quad\cos\theta<0,\quad\tan\theta<0$
Quadrant III
$\sin\theta<0,\quad\cos\theta<0,\quad\tan\theta>0$
Quadrant IV
$\sin\theta<0,\quad\cos\theta>0,\quad\tan\theta<0$
QI reference angle
$\alpha=\theta$
QII reference angle
$\alpha=\pi-\theta$
QIII reference angle
$\alpha=\theta-\pi$
QIV reference angle
$\alpha=2\pi-\theta$
45-degree triangle ratio
$1:1:\sqrt2$
30-60-90 triangle ratio
$1:\sqrt3:2$
Rationalized tangent value
$\frac{1}{\sqrt3}=\frac{\sqrt3}{3}$
Values at zero
$\sin0=0,\quad\cos0=1,\quad\tan0=0$
Values at 30 degrees
$\sin\frac{\pi}{6}=\frac12,\quad\cos\frac{\pi}{6}=\frac{\sqrt3}{2},\quad\tan\frac{\pi}{6}=\frac{\sqrt3}{3}$
Values at 45 degrees
$\sin\frac{\pi}{4}=\frac{\sqrt2}{2},\quad\cos\frac{\pi}{4}=\frac{\sqrt2}{2},\quad\tan\frac{\pi}{4}=1$
Values at 60 degrees
$\sin\frac{\pi}{3}=\frac{\sqrt3}{2},\quad\cos\frac{\pi}{3}=\frac12,\quad\tan\frac{\pi}{3}=\sqrt3$
Values at 90 degrees
$\sin\frac{\pi}{2}=1,\quad\cos\frac{\pi}{2}=0,\quad\tan\frac{\pi}{2}\text{ is undefined}$
Degree-radian equivalence
$180^\circ=\pi\text{ rad}$
Degrees to radians
$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180}$
Radians to degrees
$\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$
Coordinate recovery
$y=\pm\sqrt{1-x^2},\quad x=\pm\sqrt{1-y^2}$
Even and odd symmetry
$\cos(-\theta)=\cos\theta,\quad\sin(-\theta)=-\sin\theta,\quad\tan(-\theta)=-\tan\theta$
Sine and cosine periods
$\sin(\theta+2\pi)=\sin\theta,\quad\cos(\theta+2\pi)=\cos\theta$
Tangent period
$\tan(\theta+\pi)=\tan\theta$
Second-quadrant symmetry
$\sin(\pi-\alpha)=\sin\alpha,\quad\cos(\pi-\alpha)=-\cos\alpha$
Fourth-quadrant symmetry
$\sin(2\pi-\alpha)=-\sin\alpha,\quad\cos(2\pi-\alpha)=\cos\alpha$
Unit circle and coordinates
$x^2+y^2=1,\quad P=(\cos\theta,\sin\theta)$
Tangent
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cos\theta\ne0$
Standard sine magnitudes
$0,\ \frac12,\ \frac{\sqrt2}{2},\ \frac{\sqrt3}{2},\ 1$
Standard cosine magnitudes
$1,\ \frac{\sqrt3}{2},\ \frac{\sqrt2}{2},\ \frac12,\ 0$
Periods
$\sin(\theta+2\pi)=\sin\theta,\quad\cos(\theta+2\pi)=\cos\theta,\quad\tan(\theta+\pi)=\tan\theta$
Topic 3.6Not started
Trigonometric identities
Identities are equations that hold throughout their valid domain. Students use them to simplify expressions, prove statements and transform trigonometric equations into solvable forms.
210 min estimated
Topic 3.6 progress0%
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Learning objectives
Use the Pythagorean identity.
Use double-angle identities for sine and cosine.
Select the most useful form of cos 2θ for the expression being simplified.
Key formulas
Fundamental Pythagorean identity
$\sin^2\theta+\cos^2\theta\equiv1$
Example of an equation
$\sin\theta=\frac12$
Squared notation
$\sin^2\theta=(\sin\theta)^2,\quad\cos^2\theta=(\cos\theta)^2$
Tangent quotient identity
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cos\theta\ne0$
Reciprocal identities
$\sec\theta=\frac1{\cos\theta},\quad\csc\theta=\frac1{\sin\theta},\quad\cot\theta=\frac1{\tan\theta}$
Cotangent quotient relationship
$\cot\theta=\frac{\cos\theta}{\sin\theta},\quad\sin\theta\ne0$
Fundamental Pythagorean identity
$\sin^2\theta+\cos^2\theta=1$
Rearranged forms
$\sin^2\theta=1-\cos^2\theta,\quad\cos^2\theta=1-\sin^2\theta$
Tangent–secant identity
$1+\tan^2\theta=\sec^2\theta,\quad\cos\theta\ne0$
Cotangent–cosecant identity
$1+\cot^2\theta=\csc^2\theta,\quad\sin\theta\ne0$
Pythagorean replacements
$1-\cos^2x=\sin^2x,\quad1-\sin^2x=\cos^2x$
Reciprocal-family replacements
$\sec^2x-1=\tan^2x,\quad\csc^2x-1=\cot^2x$
Common-denominator pattern
$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd},\quad b\ne0,\ d\ne0$
Conjugate product
$(1+\cos x)(1-\cos x)=1-\cos^2x=\sin^2x$
Cosine double-angle choices
$\cos 2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$
Sine double angle
$\sin 2\theta=2\sin\theta\cos\theta$
Cosine double angle
$\cos 2\theta=\cos^2\theta-\sin^2\theta$
Alternative cosine forms
$\cos 2\theta=2\cos^2\theta-1=1-2\sin^2\theta$
Power-reduction forms
$\cos^2\theta=\frac{1+\cos 2\theta}{2},\quad\sin^2\theta=\frac{1-\cos 2\theta}{2}$
Cosine-zero angles
$\cos x=0\iff x=\frac{\pi}{2}+k\pi,\quad k\in\mathbb Z$
Sine-zero angles
$\sin x=0\iff x=k\pi,\quad k\in\mathbb Z$
Difference check
$D(x)=\mathrm{LHS}(x)-\mathrm{RHS}(x)$
Quotient identity
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cos\theta\ne0$
Reciprocal identities
$\sec\theta=\frac1{\cos\theta},\quad\csc\theta=\frac1{\sin\theta},\quad\cot\theta=\frac{\cos\theta}{\sin\theta}$
Pythagorean identity
$\sin^2\theta+\cos^2\theta=1$
Derived Pythagorean identities
$1+\tan^2\theta=\sec^2\theta,\quad1+\cot^2\theta=\csc^2\theta$
Cosine double-angle forms
$\cos 2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$
Topic 3.7Not started
Circular functions and their graphs
Students analyze sine, cosine and tangent as periodic functions. Transformations connect the basic graphs to models with different amplitude, period, phase and vertical shift.
215 min estimated
Topic 3.7 progress0%
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Learning objectives
Sketch and interpret sin x, cos x and tan x.
Identify period and amplitude where applicable.
Model periodic contexts with transformed trigonometric functions.
Key formulas
Unit-circle coordinates
$P=(\cos\theta,\sin\theta)$
Tangent quotient
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cos\theta\ne0$
Parent periods
$\sin(x+2\pi)=\sin x,\quad\cos(x+2\pi)=\cos x,\quad\tan(x+\pi)=\tan x$
Quarter-cycle spacing
$\Delta x=\frac{T}{4}$
Sine domain and range
$\operatorname{Dom}(\sin)=\mathbb R,\quad\operatorname{Ran}(\sin)=[-1,1]$
Sine zeros
$x=k\pi,\quad k\in\mathbb Z$
Sine extrema
$\sin x=1\text{ at }x=\frac{\pi}{2}+2k\pi,\quad\sin x=-1\text{ at }x=\frac{3\pi}{2}+2k\pi$
Cosine domain and range
$\operatorname{Dom}(\cos)=\mathbb R,\quad\operatorname{Ran}(\cos)=[-1,1]$
Cosine zeros
$x=\frac{\pi}{2}+k\pi,\quad k\in\mathbb Z$
Cosine extrema
$\cos x=1\text{ at }x=2k\pi,\quad\cos x=-1\text{ at }x=\pi+2k\pi$
Tangent domain
$x\in\mathbb R,\quad x\ne\frac{\pi}{2}+k\pi,\quad k\in\mathbb Z$
Tangent range
$\operatorname{Ran}(\tan)=\mathbb R$
Tangent zeros and asymptotes
$\tan x=0\text{ at }x=k\pi,\quad x=\frac{\pi}{2}+k\pi\text{ are vertical asymptotes}$
Sine and cosine range
$d-|a|\le y\le d+|a|$
Extreme values
$y_{\max}=d+|a|,\quad y_{\min}=d-|a|$
Sine and cosine amplitude
$A=|a|$
Sine and cosine period
$T=\frac{2\pi}{|b|}$
Tangent period
$T=\frac{\pi}{|b|}$
Midline
$y=d$
Transformed sine and cosine
$y=a\sin(b(x-c))+d,\quad y=a\cos(b(x-c))+d$
Transformed tangent
$y=a\tan(b(x-c))+d$
Shift from expanded inside
$bx+q=b\left(x+\frac{q}{b}\right)=b\left(x-\left(-\frac{q}{b}\right)\right)$
Tangent centre and asymptotes
$x_{\mathrm{centre}}=c+kT,\quad x_{\mathrm{asymptote}}=c+\left(k+\frac12\right)T$
Five-point x-values
$c,\quad c+\frac{T}{4},\quad c+\frac{T}{2},\quad c+\frac{3T}{4},\quad c+T$
Midline from extrema
$d=\frac{M+m}{2}$
Amplitude from extrema
$|a|=\frac{M-m}{2}$
Frequency coefficient from period
$|b|=\frac{2\pi}{T}$
Quarter-period table step
$\Delta x=\frac{T}{4}$
Sine and cosine models
$y=a\sin(b(x-c))+d,\quad y=a\cos(b(x-c))+d$
Tangent model
$y=a\tan(b(x-c))+d$
Amplitude and midline
$A=|a|,\quad y=d$
Phase shift from expanded form
$a\sin(bx+q)+d\text{ has shift }-\frac{q}{b}$
Topic 3.8Not started
Solving trigonometric equations
Trigonometric equations often have multiple solutions in an interval. Students must solve for all valid values rather than stop after the principal calculator answer.
230 min estimated
Topic 3.8 progress0%
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Learning objectives
Solve graphically and analytically over a stated finite interval.
Use exact values and identities where appropriate.
Handle equations that become quadratic in sin x, cos x or tan x.
Key formulas
Solution check
$\text{substitute each proposed }x\text{ into the original equation}$
Sine and cosine feasibility
$-1\le a\le1$
Tangent target range
$a\in\mathbb R$
Positive sine pair
$x=\alpha,\quad x=\pi-\alpha$
Positive cosine pair
$x=\alpha,\quad x=2\pi-\alpha$
Positive tangent pair
$x=\alpha,\quad x=\pi+\alpha$
Related angles in one turn
$\alpha,\quad\pi-\alpha,\quad\pi+\alpha,\quad2\pi-\alpha$
Common sine magnitudes
$0,\quad\frac12,\quad\frac{\sqrt2}{2},\quad\frac{\sqrt3}{2},\quad1$
Common tangent magnitudes
$0,\quad\frac{\sqrt3}{3},\quad1,\quad\sqrt3$
Sine second solution
$x_2=\pi-x_1\quad\text{for a positive sine value in }[0,2\pi)$
Cosine reflected solution
$x_2=2\pi-x_1\quad\text{when }x_1\in[0,\pi]$
Interval transformation
$L\le x<U\implies nL\le nx<nU\quad\text{for }n>0$
Full-turn equivalence
$360^\circ=2\pi\text{ radians}$
Degree periods
$T_{\sin}=T_{\cos}=360^\circ,\quad T_{\tan}=180^\circ$
Radian periods
$T_{\sin}=T_{\cos}=2\pi,\quad T_{\tan}=\pi$
Sine general solutions
$x=\alpha+2k\pi\quad\text{or}\quad x=\pi-\alpha+2k\pi$
Cosine general solutions
$x=\pm\alpha+2k\pi$
Tangent general solution
$x=\alpha+k\pi$
Zero-product rule
$AB=0\implies A=0\text{ or }B=0$
Pythagorean identity
$\sin^2x+\cos^2x=1$
Sine double angle
$\sin2x=2\sin x\cos x$
Cosine double angle
$\cos2x=2\cos^2x-1=1-2\sin^2x$
Intersection method
$f(x)=g(x)\iff f(x)-g(x)=0$
Quadrant-related angles
$\alpha,\quad\pi-\alpha,\quad\pi+\alpha,\quad2\pi-\alpha$
Double-angle identities
$\sin2x=2\sin x\cos x,\quad\cos2x=2\cos^2x-1=1-2\sin^2x$
Period conversion
$2\pi\text{ radians}=360^\circ,\quad\pi\text{ radians}=180^\circ$
Topic 3.9Not started
Reciprocal and inverse trigonometric functions
The reciprocal ratios sec, cosec/csc and cot extend the trigonometric toolkit and generate additional identities. Inverse trigonometric functions are treated as functions with carefully defined principal domains and ranges.
80 min estimated
Topic 3.9 progress0%
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Learning objectives
Define secant, cosecant and cotangent and connect them to sine, cosine and tangent.
Determine domains, ranges, periods, intercepts and asymptotes of reciprocal trigonometric functions.
Use reciprocal and Pythagorean identities to simplify expressions and solve equations.
Interpret arcsin, arccos and arctan as inverse functions on restricted principal branches.
Evaluate inverse trigonometric expressions exactly and numerically while respecting principal values.
Solve reciprocal and inverse trigonometric equations over stated intervals and validate every solution.
Use a GDC effectively for graphs, inverse values and numerical solutions while preserving exact answers where appropriate.
Key formulas
Reciprocal definitions
$\sec x=\frac{1}{\cos x},\qquad \csc x=\frac{1}{\sin x},\qquad \cot x=\frac{\cos x}{\sin x}$
Secant domain and range
$\operatorname{Dom}(\sec)=\mathbb R\setminus\left\{\frac\pi2+k\pi\right\},\qquad \operatorname{Ran}(\sec)=(-\infty,-1]\cup[1,\infty)$
Cosecant domain and range
$\operatorname{Dom}(\csc)=\mathbb R\setminus\{k\pi\},\qquad \operatorname{Ran}(\csc)=(-\infty,-1]\cup[1,\infty)$
Cotangent domain and range
$\operatorname{Dom}(\cot)=\mathbb R\setminus\{k\pi\},\qquad \operatorname{Ran}(\cot)=\mathbb R$
Transformed secant period
$T=\frac{2\pi}{|b|}$
Transformed secant asymptotes
$b(x-h)=\frac\pi2+k\pi$
Transformed cosecant asymptotes
$b(x-h)=k\pi$
Transformed cosecant range
$(-\infty,d-|a|]\cup[d+|a|,\infty)$
Transformed cotangent period
$T=\frac{\pi}{|b|}$
Transformed cotangent asymptotes
$b(x-h)=k\pi$
Reciprocal Pythagorean identities
$1+\tan^2x=\sec^2x,\qquad 1+\cot^2x=\csc^2x$
Inverse relation
$y=f^{-1}(x)\iff x=f(y)$
Inverse sine principal range
$\arcsin:[-1,1]\to\left[-\frac\pi2,\frac\pi2\right]$
Inverse cosine principal range
$\arccos:[-1,1]\to[0,\pi]$
Inverse tangent principal range
$\arctan:\mathbb R\to\left(-\frac\pi2,\frac\pi2\right)$
Inverse evaluation rule
$\arcsin u=\theta\iff \sin\theta=u\text{ and }-\frac\pi2\leq\theta\leq\frac\pi2$
Safe inverse compositions
$\sin(\arcsin x)=x,\quad \cos(\arccos x)=x,\quad \tan(\arctan x)=x$
Reciprocal equation conversion
$\sec u=A\Rightarrow\cos u=\frac1A,\quad \csc u=A\Rightarrow\sin u=\frac1A,\quad \cot u=A\Rightarrow\tan u=\frac1A$
Complementary inverse identity
$\arcsin x+\arccos x=\frac\pi2\qquad(-1\leq x\leq1)$
Range filter for secant and cosecant
$|\sec x|\geq1,\qquad |\csc x|\geq1$
Core reciprocal formulas
$\sec x=\frac1{\cos x},\quad \csc x=\frac1{\sin x},\quad \cot x=\frac{\cos x}{\sin x}$
Core reciprocal identities
$\sec^2x-\tan^2x=1,\qquad \csc^2x-\cot^2x=1$
Complementary inverse functions
$\arcsin x+\arccos x=\frac\pi2$
Topic 3.10Not started
Compound-angle and additional double-angle identities
Compound-angle identities express trigonometric functions of A±B in terms of functions of A and B. They are fundamental for exact values, proofs, equation solving and later calculus/series work.
80 min estimated
Topic 3.10 progress0%
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Learning objectives
Use the sine, cosine and tangent compound-angle identities with correct sign patterns.
Explain compound angles geometrically and derive the identities using rotations.
Calculate exact trigonometric values for nonstandard angles.
Use and select among the sine, cosine and tangent double-angle identities.
Derive alternative double-angle and power-reduction forms.
Simplify expressions and prove identities with clearly stated restrictions.
Solve trigonometric equations using compound-angle and double-angle identities.
Present exact and numerical solutions in an IB HL exam-ready form.
Key formulas
Sine compound-angle identities
$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$
Cosine compound-angle identities
$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$
Tangent compound-angle identities
$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$
Rotation composition
$R(A)R(B)=R(A+B)$
Exact 15-degree values
$\sin15^\circ=\frac{\sqrt6-\sqrt2}{4},\quad \cos15^\circ=\frac{\sqrt6+\sqrt2}{4},\quad \tan15^\circ=2-\sqrt3$
Sine double-angle identity
$\sin2x=2\sin x\cos x$
Cosine double-angle identity
$\cos2x=\cos^2x-\sin^2x$
Tangent double-angle identity
$\tan2x=\frac{2\tan x}{1-\tan^2x}$
Cosine double-angle family
$\cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$
Power-reduction identities
$\cos^2x=\frac{1+\cos2x}{2},\qquad \sin^2x=\frac{1-\cos2x}{2}$
Double angle in terms of tangent
$\sin2x=\frac{2t}{1+t^2},\qquad \cos2x=\frac{1-t^2}{1+t^2},\qquad t=\tan x$
Symmetric sine sum
$\sin(x+a)+\sin(x-a)=2\sin x\cos a$
Symmetric cosine sum
$\cos(x+a)+\cos(x-a)=2\cos x\cos a$
Recognition templates
$\sin x\cos a\pm\cos x\sin a=\sin(x\pm a),\qquad \cos x\cos a\mp\sin x\sin a=\cos(x\pm a)$
Compound sine general solutions
$\sin u=c\Rightarrow u=\arcsin c+2k\pi\text{ or }u=\pi-\arcsin c+2k\pi$
Double-angle equation conversion
$\cos2x=1-2\sin^2x=2\cos^2x-1$
Sine and cosine summary
$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\qquad \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$
Tangent summary
$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$
Double-angle summary
$\sin2x=2\sin x\cos x,\quad \cos2x=2\cos^2x-1=1-2\sin^2x,\quad \tan2x=\frac{2\tan x}{1-\tan^2x}$
Topic 3.11Not started
Trigonometric symmetry and graph relationships
Symmetry identities explain how the values and signs of trigonometric functions change when angles are reflected or shifted. They support exact manipulation and help students reason about whole families of solutions.
80 min estimated
Topic 3.11 progress0%
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Learning objectives
Interpret sine, cosine and tangent symmetry on the unit circle and their graphs.
Use odd/even properties and periodicity to simplify trigonometric expressions.
Apply related-angle and cofunction identities to exact values.
Connect phase shifts and reflections of sine, cosine and tangent graphs.
Determine the key features of transformed trigonometric graphs.
Generate complete trigonometric solution families using symmetry.
Use graph relationships and a GDC to solve and validate equations.
Present exact values, intervals and transformations in an IB HL exam-ready form.
Key formulas
Unit-circle coordinates
$P(\theta)=(\cos\theta,\sin\theta)$
Trig parity
$\sin(-x)=-\sin x,\qquad \cos(-x)=\cos x,\qquad \tan(-x)=-\tan x$
Basic periodicity
$\sin(x+2k\pi)=\sin x,\quad \cos(x+2k\pi)=\cos x,\quad \tan(x+k\pi)=\tan x$
Supplementary-angle identities
$\sin(\pi-\theta)=\sin\theta,\quad \cos(\pi-\theta)=-\cos\theta,\quad \tan(\pi-\theta)=-\tan\theta$
Half-turn identities
$\sin(\pi+\theta)=-\sin\theta,\quad \cos(\pi+\theta)=-\cos\theta,\quad \tan(\pi+\theta)=\tan\theta$
Once-around identities
$\sin(2\pi-\theta)=-\sin\theta,\quad \cos(2\pi-\theta)=\cos\theta,\quad \tan(2\pi-\theta)=-\tan\theta$
Cofunction identities
$\sin\left(\frac\pi2-\theta\right)=\cos\theta,\quad \cos\left(\frac\pi2-\theta\right)=\sin\theta,\quad \tan\left(\frac\pi2-\theta\right)=\cot\theta$
Sine-cosine phase relationship
$\cos x=\sin\left(x+\frac\pi2\right),\qquad \sin x=\cos\left(x-\frac\pi2\right)$
Half-period reflection
$\sin(x+\pi)=-\sin x,\qquad \cos(x+\pi)=-\cos x$
Sine-cosine transformation features
$A=|a|,\qquad T=\frac{2\pi}{|b|},\qquad \text{midline }y=d,\qquad \text{range }[d-|a|,d+|a|]$
Tangent transformation features
$T=\frac\pi{|b|},\qquad b(x-h)=\frac\pi2+k\pi$
Exact-value workflow
$\text{reduce angle}\to\text{reference angle}\to\text{quadrant sign}\to\text{exact magnitude}$
General trig solution families
$\sin x=\sin\alpha\Rightarrow x=\alpha+2k\pi\text{ or }x=\pi-\alpha+2k\pi;\quad \cos x=\cos\alpha\Rightarrow x=2k\pi\pm\alpha;\quad \tan x=\tan\alpha\Rightarrow x=\alpha+k\pi$
Parity summary
$\sin(-x)=-\sin x,\quad \cos(-x)=\cos x,\quad \tan(-x)=-\tan x$
Phase-shift summary
$\cos x=\sin\left(x+\frac\pi2\right),\qquad \sin x=\cos\left(x-\frac\pi2\right)$
Solution-family summary
$\sin x=\sin\alpha\Rightarrow x=\alpha+2k\pi\text{ or }\pi-\alpha+2k\pi;\quad \cos x=\cos\alpha\Rightarrow x=2k\pi\pm\alpha;\quad \tan x=\tan\alpha\Rightarrow x=\alpha+k\pi$
Topic 3.12Not started
Vectors: components, magnitude and geometry
Vectors provide an algebraic language for position, displacement and direction in two and three dimensions. At HL they become a major tool for proving geometric relationships and describing lines and planes.
80 min estimated
Topic 3.12 progress0%
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Learning objectives
Distinguish vectors from scalars and use standard vector notation confidently.
Represent vectors by components in two and three dimensions and by the basis vectors i, j and k.
Use position vectors and displacement vectors to connect coordinates with geometry.
Calculate vector magnitudes and construct unit vectors in specified directions.
Perform vector addition, subtraction and scalar multiplication algebraically and geometrically.
Find a vector between two points and use vector equations to determine unknown coordinates.
Test whether vectors are parallel and whether points are collinear.
Present complete, validated vector solutions in IB Mathematics AA HL problems.
Key formulas
2D component form
$\mathbf{v}=\begin{pmatrix}v_1\\[2pt]v_2\end{pmatrix}$
3D component form
$\mathbf{v}=\begin{pmatrix}v_1\\[2pt]v_2\\[2pt]v_3\end{pmatrix}$
Reversing a directed vector
$\overrightarrow{BA}=-\overrightarrow{AB}$
Standard basis decomposition
$\begin{pmatrix}a\\[2pt]b\\[2pt]c\end{pmatrix}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$
Position vector
$P(x,y,z)\Longleftrightarrow\overrightarrow{OP}=\begin{pmatrix}x\\[2pt]y\\[2pt]z\end{pmatrix}$
2D magnitude
$\left\lVert\begin{pmatrix}a\\[2pt]b\end{pmatrix}\right\rVert=\sqrt{a^2+b^2}$
3D magnitude
$\left\lVert\begin{pmatrix}a\\[2pt]b\\[2pt]c\end{pmatrix}\right\rVert=\sqrt{a^2+b^2+c^2}$
Distance between points
$AB=\lVert\overrightarrow{AB}\rVert$
Unit vector
$\widehat{\mathbf{v}}=\frac{\mathbf{v}}{\lVert\mathbf{v}\rVert},\quad \mathbf{v}\ne\mathbf{0}$
Vector with prescribed magnitude
$\mathbf{w}=M\frac{\mathbf{v}}{\lVert\mathbf{v}\rVert},\quad M>0$
Componentwise addition
$\begin{pmatrix}a\\[2pt]b\\[2pt]c\end{pmatrix}+\begin{pmatrix}p\\[2pt]q\\[2pt]r\end{pmatrix}=\begin{pmatrix}a+p\\[2pt]b+q\\[2pt]c+r\end{pmatrix}$
Vector subtraction
$\mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v})$
Scalar multiplication
$k\begin{pmatrix}a\\[2pt]b\\[2pt]c\end{pmatrix}=\begin{pmatrix}ka\\[2pt]kb\\[2pt]kc\end{pmatrix}$
Magnitude of a scalar multiple
$\lVert k\mathbf{v}\rVert=|k|\lVert\mathbf{v}\rVert$
Vector between points
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$
Triangle law
$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$
Recover an endpoint
$\overrightarrow{OB}=\overrightarrow{OA}+\overrightarrow{AB}$
Midpoint position vector
$\mathbf{m}=\frac{\mathbf{a}+\mathbf{b}}{2}$
Point on a segment or line
$\mathbf{p}=(1-t)\mathbf{a}+t\mathbf{b}=\mathbf{a}+t(\mathbf{b}-\mathbf{a})$
Parallelogram fourth vertex
$\mathbf{c}=\mathbf{b}+\mathbf{d}-\mathbf{a}$
Parallel-vector test
$\mathbf{u}\parallel\mathbf{v}\Longleftrightarrow\mathbf{u}=\lambda\mathbf{v}\quad(\mathbf{u},\mathbf{v}\ne\mathbf{0})$
Collinearity test
$A,B,C\text{ collinear}\Longleftrightarrow\overrightarrow{AB}=\lambda\overrightarrow{AC}\text{ for some }\lambda\in\mathbb{R}$
Equality of vectors
$\begin{pmatrix}a\\[2pt]b\\[2pt]c\end{pmatrix}=\begin{pmatrix}p\\[2pt]q\\[2pt]r\end{pmatrix}\Longleftrightarrow a=p,\ b=q,\ c=r$
Resultant displacement
$\mathbf{r}=\mathbf{d}_1+\mathbf{d}_2+\cdots+\mathbf{d}_n$
Core component model
$\mathbf{v}=\begin{pmatrix}v_1\\[2pt]v_2\\[2pt]v_3\end{pmatrix}=v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}$
Core length model
$\lVert\mathbf{v}\rVert=\sqrt{v_1^2+v_2^2+v_3^2}$
Core direction model
$\widehat{\mathbf{v}}=\frac{\mathbf{v}}{\lVert\mathbf{v}\rVert}$
Core displacement model
$\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$
Core geometry model
$\mathbf{p}=\mathbf{a}+t(\mathbf{b}-\mathbf{a})$
Topic 3.13Not started
Scalar (dot) product
The scalar product connects algebraic components with geometric angle. It provides a fast test for perpendicularity and is used repeatedly when finding angles between lines and planes.
80 min estimated
Topic 3.13 progress0%
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Learning objectives
Calculate scalar products from vector components in two and three dimensions.
Connect the component definition of the dot product to magnitudes and the included angle.
Find exact and approximate angles between nonzero vectors.
Test perpendicularity and use orthogonality to find unknown components.
Interpret the sign of a dot product geometrically.
Find scalar and vector projections in geometric and contextual problems.
Use the dot product in coordinate-geometry proofs and applications.
Present complete, validated IB Mathematics AA HL solutions using exact values appropriately.
Key formulas
2D component dot product
$\begin{pmatrix}a_1\\[2pt]a_2\end{pmatrix}\cdot\begin{pmatrix}b_1\\[2pt]b_2\end{pmatrix}=a_1b_1+a_2b_2$
3D component dot product
$\begin{pmatrix}a_1\\[2pt]a_2\\[2pt]a_3\end{pmatrix}\cdot\begin{pmatrix}b_1\\[2pt]b_2\\[2pt]b_3\end{pmatrix}=a_1b_1+a_2b_2+a_3b_3$
Geometric dot product
$\mathbf{a}\cdot\mathbf{b}=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\cos\theta$
Dot-product bound
$|\mathbf{a}\cdot\mathbf{b}|\le\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert$
Angle between nonzero vectors
$\theta=\cos^{-1}\left(\frac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert}\right)$
Sign and angle
$\mathbf{a}\cdot\mathbf{b}\begin{cases}>0,&\theta<\pi/2\\[2pt]=0,&\theta=\pi/2\\[2pt]<0,&\theta>\pi/2\end{cases}$
Orthogonality criterion
$\mathbf{a}\perp\mathbf{b}\Longleftrightarrow\mathbf{a}\cdot\mathbf{b}=0\quad(\mathbf{a},\mathbf{b}\ne\mathbf{0})$
Perpendicular-component equation
$a_1b_1+a_2b_2+a_3b_3=0$
Scalar projection
$\operatorname{comp}_{\mathbf{b}}\mathbf{a}=\frac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{b}\rVert}$
Vector projection
$\operatorname{proj}_{\mathbf{b}}\mathbf{a}=\frac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{b}\rVert^2}\mathbf{b}$
Perpendicular remainder
$\mathbf{a}_{\perp}=\mathbf{a}-\operatorname{proj}_{\mathbf{b}}\mathbf{a}$
Commutativity
$\mathbf{a}\cdot\mathbf{b}=\mathbf{b}\cdot\mathbf{a}$
Distributivity
$\mathbf{a}\cdot(\mathbf{b}+\mathbf{c})=\mathbf{a}\cdot\mathbf{b}+\mathbf{a}\cdot\mathbf{c}$
Scalar compatibility
$(k\mathbf{a})\cdot\mathbf{b}=k(\mathbf{a}\cdot\mathbf{b})$
Self-dot product
$\mathbf{a}\cdot\mathbf{a}=\lVert\mathbf{a}\rVert^2$
Magnitude of a sum
$\lVert\mathbf{a}+\mathbf{b}\rVert^2=\lVert\mathbf{a}\rVert^2+2\mathbf{a}\cdot\mathbf{b}+\lVert\mathbf{b}\rVert^2$
Magnitude of a difference
$\lVert\mathbf{a}-\mathbf{b}\rVert^2=\lVert\mathbf{a}\rVert^2-2\mathbf{a}\cdot\mathbf{b}+\lVert\mathbf{b}\rVert^2$
Parallelogram law
$\lVert\mathbf{a}+\mathbf{b}\rVert^2+\lVert\mathbf{a}-\mathbf{b}\rVert^2=2\lVert\mathbf{a}\rVert^2+2\lVert\mathbf{b}\rVert^2$
Vertex-angle model
$\cos\angle ABC=\frac{\overrightarrow{BA}\cdot\overrightarrow{BC}}{\lVert\overrightarrow{BA}\rVert\lVert\overrightarrow{BC}\rVert}$
Work done
$W=\mathbf{F}\cdot\mathbf{s}$
Resolved scalar component
$\operatorname{comp}_{\mathbf{d}}\mathbf{v}=\frac{\mathbf{v}\cdot\mathbf{d}}{\lVert\mathbf{d}\rVert}$
Core component formula
$\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2+a_3b_3$
Core geometric formula
$\mathbf{a}\cdot\mathbf{b}=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\cos\theta$
Core angle formula
$\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert}$
Core orthogonality formula
$\mathbf{a}\perp\mathbf{b}\Longleftrightarrow\mathbf{a}\cdot\mathbf{b}=0$
Core vector projection formula
$\operatorname{proj}_{\mathbf{b}}\mathbf{a}=\frac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{b}\rVert^2}\mathbf{b}$
Topic 3.14Not started
Vector equation of a line
A line in two or three dimensions is described by one fixed point and a nonzero direction vector. Vector and parametric forms make points, intersections and motion along the line easy to analyze.
80 min estimated
Topic 3.14 progress0%
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Learning objectives
Distinguish position vectors from direction vectors in two and three dimensions.
Write and interpret the vector equation of a line.
Convert accurately between vector and parametric equations.
Construct a line from a point and direction or from two distinct points.
Determine exactly whether a point lies on a line.
Recognize equivalent equations of the same line.
Use line parameters in coordinate-geometry and motion contexts.
Present complete, validated IB Mathematics AA HL solutions using correct vector notation.
Key formulas
Position vector in 3D
$\overrightarrow{OP}=\begin{pmatrix}x\\[2pt]y\\[2pt]z\end{pmatrix}$
Vector between points
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$
Vector equation of a line
$\mathbf{r}=\mathbf{a}+t\mathbf{d},\qquad t\in\mathbb{R},\ \mathbf{d}\ne\mathbf{0}$
Parametric equations in 3D
$x=x_0+lt,\qquad y=y_0+mt,\qquad z=z_0+nt$
Line through two points
$\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{a}),\qquad t\in\mathbb{R}$
Point-on-line criterion
$\mathbf{q}-\mathbf{a}=t\mathbf{d}\text{ for one }t\in\mathbb{R}$
Symmetric form
$\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$
Same-line test
$\mathbf{d}_2=k\mathbf{d}_1\ (k\ne0)\quad\text{and}\quad \mathbf{a}_2-\mathbf{a}_1=\lambda\mathbf{d}_1$
Line segment
$\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{a}),\qquad 0\le t\le1$
Constant-velocity line
$\mathbf{r}(t)=\mathbf{r}_0+t\mathbf{v}$
Core vector form
$\mathbf{r}=\mathbf{a}+t\mathbf{d}$
Core parametric form
$x=x_0+lt,\quad y=y_0+mt,\quad z=z_0+nt$
Core two-point form
$\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{a})$
Core point test
$\mathbf{q}-\mathbf{a}=t\mathbf{d}$
Topic 3.15Not started
Relationships between lines in space
Classify two lines in three-dimensional space as coincident, parallel, intersecting, perpendicular or skew by comparing direction vectors and solving parameter equations. Find intersections and acute angles with exact vector methods.
85 min estimated
Topic 3.15 progress0%
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Learning objectives
Represent two 3D lines with independent parameters and compare their direction vectors.
Distinguish parallel, coincident, intersecting, perpendicular and skew lines using complete algebraic tests.
Solve and verify simultaneous parameter equations to find a line intersection point.
Use the scalar product to find the acute angle between two lines and test perpendicularity correctly.
Interpret line relationships geometrically and communicate exact IB HL reasoning.
Use a GDC to support calculations while retaining algebraic justification and validation.
Recognize common logical errors, including treating every nonparallel pair as intersecting.
Solve multi-step and parameter-based line-relationship problems in two and three dimensions.
Key formulas
General pair of 3D lines
$L_1:\mathbf r=\mathbf a+\lambda\mathbf u,\qquad L_2:\mathbf r=\mathbf b+\mu\mathbf v$
Parallel-direction test
$\mathbf v=k\mathbf u\quad\text{for some }k\ne0$
Distinct-parallel test
$\mathbf v=k\mathbf u\quad\text{and}\quad \mathbf b-\mathbf a\ne t\mathbf u\ \text{for every }t\in\mathbb R$
Coincident-line test
$\mathbf v=k\mathbf u\quad\text{and}\quad\mathbf b-\mathbf a=t\mathbf u$
Line-intersection system
$\mathbf a+\lambda\mathbf u=\mathbf b+\mu\mathbf v$
Perpendicular-line criterion
$\mathbf u\cdot\mathbf v=0\quad\text{and the lines intersect}$
Skew-line classification
$\mathbf v\ne k\mathbf u\ \text{for every }k\quad\text{and}\quad\mathbf a+\lambda\mathbf u=\mathbf b+\mu\mathbf v\ \text{has no solution}$
Classification decision
$\begin{cases}\mathbf v=k\mathbf u:\ \text{parallel or coincident},\\[2pt]\mathbf v\ne k\mathbf u:\ \text{intersecting or skew}.\end{cases}$
Angle between lines
$\cos\theta=\frac{|\mathbf u\cdot\mathbf v|}{\lVert\mathbf u\rVert\lVert\mathbf v\rVert},\qquad 0\le\theta\le\frac\pi2$
Parallel-direction test
$\mathbf v=k\mathbf u,\quad k\ne0$
Intersection test
$\mathbf a+\lambda\mathbf u=\mathbf b+\mu\mathbf v$
Perpendicular-line test
$\mathbf u\cdot\mathbf v=0\quad\text{and the lines intersect}$
Acute line angle
$\cos\theta=\frac{|\mathbf u\cdot\mathbf v|}{\lVert\mathbf u\rVert\lVert\mathbf v\rVert}$
Topic 3.16Not started
Vector (cross) product
Compute and interpret the three-dimensional cross product. Use its direction for oriented normal vectors and its magnitude for angles, parallel tests, areas, plane geometry and volume applications.
90 min estimated
Topic 3.16 progress0%
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Learning objectives
Define the cross product of two 3D vectors and distinguish it from the scalar product.
Compute a cross product accurately from components or determinant notation.
Use the right-hand rule and order of multiplication to determine its direction.
Connect cross-product magnitude with the included angle and test for parallel vectors.
Find vectors perpendicular to two given vectors and verify orthogonality.
Calculate exact parallelogram and triangle areas from vectors or coordinate points.
Apply cross products to normal vectors, plane equations and three-dimensional volumes.
Use exact IB HL reasoning, calculator checks and appropriate geometric interpretation.
Key formulas
Cross-product orthogonality
$(\mathbf a\times\mathbf b)\cdot\mathbf a=0,\qquad(\mathbf a\times\mathbf b)\cdot\mathbf b=0$
Cross product in components
$\mathbf a\times\mathbf b=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1)$
Determinant form
$\mathbf a\times\mathbf b=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}$
Anti-commutativity
$\mathbf b\times\mathbf a=-(\mathbf a\times\mathbf b)$
Basis-vector cycle
$\mathbf i\times\mathbf j=\mathbf k,\quad\mathbf j\times\mathbf k=\mathbf i,\quad\mathbf k\times\mathbf i=\mathbf j$
Cross-product magnitude
$\lVert\mathbf a\times\mathbf b\rVert=\lVert\mathbf a\rVert\lVert\mathbf b\rVert\sin\theta$
Dot-cross magnitude identity
$(\mathbf a\cdot\mathbf b)^2+\lVert\mathbf a\times\mathbf b\rVert^2=\lVert\mathbf a\rVert^2\lVert\mathbf b\rVert^2$
Oriented unit normal
$\widehat{\mathbf n}=\frac{\mathbf a\times\mathbf b}{\lVert\mathbf a\times\mathbf b\rVert}$
Parallelogram area
$A_{\mathrm{parallelogram}}=\lVert\mathbf a\times\mathbf b\rVert$
Triangle area
$A_{\triangle ABC}=\frac12\lVert\overrightarrow{AB}\times\overrightarrow{AC}\rVert$
Parallel-vector test
$\mathbf a\times\mathbf b=\mathbf0\quad\Longleftrightarrow\quad\mathbf a\parallel\mathbf b$
Collinearity test
$\overrightarrow{AB}\times\overrightarrow{AC}=\mathbf0$
Normal from spanning vectors
$\mathbf n=\mathbf u\times\mathbf v$
Plane through a point
$\mathbf n\cdot(\mathbf r-\mathbf a)=0$
Parallelepiped volume
$V=|\mathbf a\cdot(\mathbf b\times\mathbf c)|$
Tetrahedron volume
$V_{\mathrm{tetrahedron}}=\frac16|\mathbf a\cdot(\mathbf b\times\mathbf c)|$
Coplanarity test
$\mathbf a\cdot(\mathbf b\times\mathbf c)=0$
Distributive law
$\mathbf a\times(\mathbf b+\mathbf c)=\mathbf a\times\mathbf b+\mathbf a\times\mathbf c$
Scalar compatibility
$(k\mathbf a)\times\mathbf b=k(\mathbf a\times\mathbf b)=\mathbf a\times(k\mathbf b)$
Self cross product
$\mathbf a\times\mathbf a=\mathbf0$
Component formula
$\mathbf a\times\mathbf b=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1)$
Right-hand orientation
$\mathbf b\times\mathbf a=-(\mathbf a\times\mathbf b)$
Magnitude
$\lVert\mathbf a\times\mathbf b\rVert=\lVert\mathbf a\rVert\lVert\mathbf b\rVert\sin\theta$
Parallelogram and triangle areas
$A_P=\lVert\mathbf a\times\mathbf b\rVert,\qquad A_T=\frac12\lVert\mathbf a\times\mathbf b\rVert$
Parallel test
$\mathbf a\times\mathbf b=\mathbf0$
Volume
$V=|\mathbf a\cdot(\mathbf b\times\mathbf c)|$
Topic 3.17Not started
Vector equations of a plane
Represent planes in three-dimensional space using vector, parametric, normal and Cartesian equations. Construct planes from geometric information, convert between forms and test point membership with exact vector methods.
90 min estimated
Topic 3.17 progress0%
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Learning objectives
Interpret a plane as an anchor point plus two independent in-plane directions.
Write and use vector and parametric equations of a plane.
Find and interpret a normal vector using a cross product or Cartesian coefficients.
Construct a plane from a point and normal, three noncollinear points, or a line and an external point.
Convert accurately between vector, parametric, normal and Cartesian forms.
Test whether a point lies on a plane using either parameters or substitution.
Recognize equivalent plane equations and invalid dependent direction choices.
Present exact IB HL working with appropriate calculator verification and geometric conclusions.
Key formulas
Independent-direction test
$\mathbf u\times\mathbf v\ne\mathbf0$
Vector equation of a plane
$\mathbf r=\mathbf a+\lambda\mathbf u+\mu\mathbf v,\qquad \lambda,\mu\in\mathbb R$
Parametric plane equations
$x=x_0+\lambda u_1+\mu v_1,\quad y=y_0+\lambda u_2+\mu v_2,\quad z=z_0+\lambda u_3+\mu v_3$
Degenerate direction condition
$\mathbf u\times\mathbf v=\mathbf0\quad\Longrightarrow\quad\text{no plane is spanned}$
Normal from two directions
$\mathbf n=\mathbf u\times\mathbf v$
Normal orthogonality
$\mathbf n\cdot\mathbf u=0,\qquad\mathbf n\cdot\mathbf v=0$
Normal form of a plane
$\mathbf n\cdot(\mathbf r-\mathbf a)=0$
Scalar form of a plane
$\mathbf r\cdot\mathbf n=\mathbf a\cdot\mathbf n$
Cartesian plane
$Ax+By+Cz=D,\qquad\mathbf n=(A,B,C)$
Point-normal construction
$\mathbf n\cdot(\mathbf r-\mathbf p)=0$
Plane through three points
$\mathbf n=\overrightarrow{AB}\times\overrightarrow{AC},\qquad \mathbf n\cdot(\mathbf r-\mathbf a)=0$
Vector-to-Cartesian conversion
$\mathbf n=\mathbf u\times\mathbf v,\qquad \mathbf n\cdot(\mathbf r-\mathbf a)=0$
Direction condition from Cartesian form
$A u_1+B u_2+C u_3=0$
Cartesian membership test
$P=(x_1,y_1,z_1)\text{ lies on }Ax+By+Cz=D\iff Ax_1+By_1+Cz_1=D$
Plane containing line and point
$\mathbf n=\mathbf d\times(\mathbf q-\mathbf a),\qquad \mathbf n\cdot(\mathbf r-\mathbf a)=0$
Vector plane
$\mathbf r=\mathbf a+\lambda\mathbf u+\mu\mathbf v$
Parametric plane
$x=x_0+\lambda u_1+\mu v_1,\quad y=y_0+\lambda u_2+\mu v_2,\quad z=z_0+\lambda u_3+\mu v_3$
Normal from directions
$\mathbf n=\mathbf u\times\mathbf v$
Normal form
$\mathbf n\cdot(\mathbf r-\mathbf a)=0$
Cartesian form
$Ax+By+Cz=D,\qquad\mathbf n=(A,B,C)$
Three-point plane
$\mathbf n=\overrightarrow{AB}\times\overrightarrow{AC}$
Topic 3.18Not started
Intersections and angles involving lines and planes
Combine vector equations, normal vectors, dot products and linear systems to classify intersections and calculate acute angles between lines and planes in three dimensions.
105 min estimated
Topic 3.18 progress0%
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Learning objectives
Classify and find intersections between a line and a plane, including parallel and contained cases.
Find and interpret the line of intersection of two non-parallel planes.
Solve systems representing three planes and distinguish a unique point, a common line, no common point and coincident families.
Test parallel and perpendicular relationships using direction and normal vectors.
Calculate acute angles between two lines, a line and a plane, and two planes.
Use vector and parametric forms together with Cartesian equations in multi-step three-dimensional problems.
Use a GDC to solve systems and verify vector calculations while showing sufficient mathematical method.
Communicate IB HL solutions with correct parameters, angle conventions, exact values and geometric conclusions.
Key formulas
Vector equation of a line
$\mathbf r=\mathbf a+t\mathbf d,\quad t\in\mathbb R$
Cartesian equation of a plane
$\mathbf n\cdot\mathbf r=c\quad\Longleftrightarrow\quad Ax+By+Cz=D$
Orthogonality test
$\mathbf u\cdot\mathbf v=0$
Common direction of two planes
$\mathbf d=\mathbf n_1\times\mathbf n_2$
Line-plane substitution
$\mathbf n\cdot(\mathbf a+t\mathbf d)=c$
Unique line-plane intersection
$t=\frac{c-\mathbf n\cdot\mathbf a}{\mathbf n\cdot\mathbf d}\quad\text{when }\mathbf n\cdot\mathbf d\ne0$
Parallel line-plane test
$\mathbf d\cdot\mathbf n=0$
Contained-line test
$\mathbf d\cdot\mathbf n=0\quad\text{and}\quad\mathbf n\cdot\mathbf a=c$
Plane-intersection direction
$\mathbf d=\mathbf n_1\times\mathbf n_2$
Intersection-line checks
$\mathbf d\cdot\mathbf n_1=0\quad\text{and}\quad\mathbf d\cdot\mathbf n_2=0$
Line from anchor and common direction
$\mathbf r=\mathbf p+t(\mathbf n_1\times\mathbf n_2)$
Parallel planes
$\mathbf n_1=k\mathbf n_2\quad(k\ne0)$
Coincident Cartesian planes
$(A_1,B_1,C_1,D_1)=k(A_2,B_2,C_2,D_2)$
Three-plane system
$\begin{bmatrix}A_1&B_1&C_1\\A_2&B_2&C_2\\A_3&B_3&C_3\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}D_1\\D_2\\D_3\end{bmatrix}$
Unique three-plane point
$\det\!\begin{bmatrix}A_1&B_1&C_1\\A_2&B_2&C_2\\A_3&B_3&C_3\end{bmatrix}\ne0$
Parallel lines
$\mathbf d_1=k\mathbf d_2$
Perpendicular intersecting lines
$\mathbf d_1\cdot\mathbf d_2=0\quad\text{and the lines intersect}$
Line parallel to plane
$\mathbf d\cdot\mathbf n=0$
Line perpendicular to plane
$\mathbf d=k\mathbf n$
Perpendicular planes
$\mathbf n_1\cdot\mathbf n_2=0$
Line-line angle
$\cos\theta=\frac{|\mathbf d_1\cdot\mathbf d_2|}{\lVert\mathbf d_1\rVert\lVert\mathbf d_2\rVert},\quad0\le\theta\le\frac\pi2$
Line-plane angle
$\sin\phi=\frac{|\mathbf d\cdot\mathbf n|}{\lVert\mathbf d\rVert\lVert\mathbf n\rVert},\quad0\le\phi\le\frac\pi2$
Plane-plane angle
$\cos\theta=\frac{|\mathbf n_1\cdot\mathbf n_2|}{\lVert\mathbf n_1\rVert\lVert\mathbf n_2\rVert},\quad0\le\theta\le\frac\pi2$
Augmented row contradiction
$[\,0\quad0\quad0\mid k\,],\quad k\ne0$
Matrix solution when invertible
$\mathbf x=A^{-1}\mathbf b\quad\text{if }\det A\ne0$
Line-plane intersection equation
$\mathbf n\cdot(\mathbf a+t\mathbf d)=c$
Two-plane intersection direction
$\mathbf d=\mathbf n_1\times\mathbf n_2$
Line-line acute angle
$\cos\theta=\frac{|\mathbf d_1\cdot\mathbf d_2|}{\lVert\mathbf d_1\rVert\lVert\mathbf d_2\rVert}$
Line-plane acute angle
$\sin\phi=\frac{|\mathbf d\cdot\mathbf n|}{\lVert\mathbf d\rVert\lVert\mathbf n\rVert}$
Plane-plane acute angle
$\cos\theta=\frac{|\mathbf n_1\cdot\mathbf n_2|}{\lVert\mathbf n_1\rVert\lVert\mathbf n_2\rVert}$
Unit 4
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Statistics and Probability
AA HL statistics and probability begins with the same data, correlation, probability and distribution content studied at SL, then extends conditional probability with Bayes’ theorem and develops the theory of discrete and continuous random variables in greater depth.
14 topics
Unit 4 progress0%
Topics in this unit
Topic 4.1Not started
Data concepts, sampling and reliability
Statistics begins with the quality of the data. Students need to distinguish populations and samples, types of variables and sources of bias before applying numerical techniques.
150 min estimated
Topic 4.1 progress0%
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Learning objectives
Distinguish population, sample and random sample.
Classify data as discrete or continuous where relevant.
Recognize biased sampling, questionable sources and the effect of outliers.
Evaluate whether a dataset can support the intended conclusion.
Key formulas
Sample proportion
$\hat p=\frac{x}{n}$
Sampling fraction
$f=\frac{n}{N}$
Systematic sampling interval
$k=\frac{N}{n}$
Selected positions
$r,r+k,r+2k,\ldots,r+(n-1)k$
Proportional stratum allocation
$n_h=n\frac{N_h}{N}$
Sampling calculation checklist
$\text{define population}\to\text{choose frame}\to\text{select method}\to\text{evaluate limitations}$
Core sampling formulas
$\hat p=\frac{x}{n},\qquad k=\frac{N}{n},\qquad n_h=n\frac{N_h}{N}$
Systematic interval
$k=\frac{N}{n}$
Stratified allocation
$n_h=n\frac{N_h}{N}$
Topic 4.2Not started
Presentation of data
Data displays should reveal distribution, center, spread and unusual values. Students should be able to read a graph critically rather than only construct it.
165 min estimated
Topic 4.2 progress0%
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Learning objectives
Interpret frequency distributions and histograms.
Use cumulative-frequency graphs.
Estimate median, quartiles and percentiles from cumulative information.
Interpret box-and-whisker plots and compare distributions.
Key formulas
Relative frequency
$\text{relative frequency}=\frac{f}{n}$
Percentage frequency
$\text{percentage frequency}=\frac{f}{n}\times100\%$
Cumulative frequency
$F_i=\sum\limits_{j=1}^{i}f_j$
Class width
$w=U-L$
Class midpoint
$m=\frac{L+U}{2}$
Frequency density
$\text{frequency density}=\frac{f}{w}$
Histogram area
$f=(\text{frequency density})(\text{class width})$
Frequency between two values
$f(a<x\le b)\approx F(b)-F(a)$
Quartile levels
$Q_1:\frac{N}{4},\qquad Q_2:\frac{N}{2},\qquad Q_3:\frac{3N}{4}$
Percentile level
$\text{CF level}=\frac{p}{100}N$
Interquartile range
$\operatorname{IQR}=Q_3-Q_1$
Range
$\text{range}=\max-\min$
Outlier fences
$Q_1-1.5\operatorname{IQR},\qquad Q_3+1.5\operatorname{IQR}$
Difference in medians
$\Delta\widetilde{x}=\widetilde{x}_A-\widetilde{x}_B$
Display workflow
$\text{classify data}\to\text{calculate}\to\text{construct}\to\text{interpret}$
Core presentation formulas
$\frac{f}{n},\qquad \frac{f}{w},\qquad \operatorname{IQR}=Q_3-Q_1$
Relative frequency
$\frac{f}{n}$
Frequency density
$\frac{f}{w}$
Quartile levels
$\frac{N}{4},\quad\frac{N}{2},\quad\frac{3N}{4}$
Topic 4.3Not started
Measures of center and spread
A useful summary of data normally needs both a measure of center and a measure of variability. AA HL students should also understand how transformations of every data value affect these statistics.
170 min estimated
Topic 4.3 progress0%
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Learning objectives
Calculate or interpret mean, median and mode.
Estimate a mean from grouped data and identify the modal class.
Use range, interquartile range, variance and standard deviation.
Determine the effect of adding or multiplying all data values by a constant.
Key formulas
Sample mean
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
Zero-sum deviations
$\sum\limits_{i=1}^{n}(x_i-\bar{x})=0$
Odd-n median position
$\frac{n+1}{2}$
Even-n median
$\widetilde{x}=\frac{x_{(\frac{n}{2})}+x_{(\frac{n}{2}+1)}}{2}$
Quartile cumulative levels
$\frac{N}{4},\qquad\frac{N}{2},\qquad\frac{3N}{4}$
Percentile cumulative level
$\frac{p}{100}N$
Range
$\text{range}=x_{\max}-x_{\min}$
Interquartile range
$\operatorname{IQR}=Q_3-Q_1$
Population variance
$\sigma^2=\frac{1}{N}\sum\limits_{i=1}^{N}(x_i-\mu)^2$
Population computational form
$\sigma^2=\frac{1}{N}\sum\limits_{i=1}^{N}x_i^2-\mu^2$
Sample variance
$s^2=\frac{1}{n-1}\sum\limits_{i=1}^{n}(x_i-\bar{x})^2$
Standard deviations
$\sigma=\sqrt{\sigma^2},\qquad s=\sqrt{s^2}$
Frequency mean
$\bar{x}=\frac{\sum\limits_{i=1}^{k}f_ix_i}{\sum\limits_{i=1}^{k}f_i}$
Grouped mean estimate
$\bar{x}\approx\frac{\sum\limits_{i=1}^{k}f_im_i}{\sum\limits_{i=1}^{k}f_i}$
Grouped population variance estimate
$\sigma^2\approx\frac{\sum\limits_{i=1}^{k}f_im_i^2}{\sum\limits_{i=1}^{k}f_i}-\bar{x}^{\,2}$
Transformed mean
$\bar{y}=a\bar{x}+b$
Transformed variance
$\operatorname{Var}(Y)=a^2\operatorname{Var}(X)$
Transformed standard deviation
$\sigma_Y=|a|\sigma_X$
Transformed IQR
$\operatorname{IQR}_Y=|a|\operatorname{IQR}_X$
Common potential-outlier fences
$Q_1-1.5\operatorname{IQR},\qquad Q_3+1.5\operatorname{IQR}$
Exam workflow
$\text{identify data role}\to\text{calculate}\to\text{interpret}\to\text{check units}$
Core center-and-spread formulas
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i,\qquad \operatorname{IQR}=Q_3-Q_1,\qquad \sigma=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}(x_i-\mu)^2}$
Mean
$\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}x_i$
IQR
$\operatorname{IQR}=Q_3-Q_1$
Topic 4.4Not started
Bivariate data and linear correlation (current syllabus)
Bivariate analysis studies the relationship between two variables. Correlation describes strength and direction of linear association; regression provides a line for prediction. Neither automatically establishes causation.
175 min estimated
Topic 4.4 progress0%
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Learning objectives
Interpret scatter plots and Pearson’s correlation coefficient r.
Draw or interpret a line of best fit and the mean point.
Use the regression line of y on x for prediction when x is the explanatory variable.
Interpret slope and intercept in context and avoid unjustified extrapolation.
Key formulas
Pearson’s product-moment correlation coefficient
$r=\frac{\displaystyle\sum\limits_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\displaystyle\left[\sum\limits_{i=1}^{n}(x_i-\bar{x})^2\right]\left[\sum\limits_{i=1}^{n}(y_i-\bar{y})^2\right]}}$
Range of r
$-1\le r\le1$
Strength comparison
$|r_1|>|r_2|\quad\Longrightarrow\quad r_1\text{ represents the stronger linear association}$
Mean point
$(\bar{x},\bar{y})=\left(\frac{1}{n}\sum\limits_{i=1}^{n}x_i,\frac{1}{n}\sum\limits_{i=1}^{n}y_i\right)$
Gradient from two points on a line
$m=\frac{y_2-y_1}{x_2-x_1}$
Regression line of y on x
$\hat{y}=ax+b$
Regression coefficients
$a=r\frac{s_y}{s_x},\qquad b=\bar{y}-a\bar{x}$
Residual
$e=y-\hat{y}$
Correlation and regression toolkit
$-1\le r\le1,\qquad \hat{y}=ax+b,\qquad e=y-\hat{y}$
Pearson correlation coefficient
$r=\frac{\displaystyle\sum\limits_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\displaystyle\left[\sum\limits_{i=1}^{n}(x_i-\bar{x})^2\right]\left[\sum\limits_{i=1}^{n}(y_i-\bar{y})^2\right]}}$
Regression and residual
$\hat{y}=ax+b,\qquad e=y-\hat{y}$
Topic 4.5Not started
Probability concepts
Probability formalizes uncertainty through outcomes, events and sample spaces. Students should move between theoretical probability, relative frequency and expected counts.
165 min estimated
Topic 4.5 progress0%
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Learning objectives
Use sample-space notation and identify events.
Calculate probability from equally likely outcomes.
Use complements and expected number of occurrences.
Key formulas
Counting outcomes in successive choices
$n(U)=n_1n_2\cdots n_k$
Event containment
$A\subseteq U$
Probability bounds
$0\le P(A)\le1$
Impossible and certain events
$P(\varnothing)=0,\qquad P(U)=1$
Equally likely outcomes
$P(A)=\frac{n(A)}{n(U)}$
Complement rule
$P(A')=1-P(A)$
Complement total
$P(A)+P(A')=1$
Mutually exclusive condition
$A\cap B=\varnothing\qquad\Longleftrightarrow\qquad P(A\cap B)=0$
Addition rule for mutually exclusive events
$P(A\cup B)=P(A)+P(B)$
Relative frequency
$\widehat{P}(A)=\frac{f_A}{N}$
Expected number of occurrences
$E(\text{number of occurrences of }A)=NP(A)$
Total probability for exhaustive outcomes
$\sum\limits_{i=1}^{k}P(A_i)=1$
Core rules
$P(A')=1-P(A),\qquad P(A\cup B)=P(A)+P(B)\ \text{if }A\cap B=\varnothing$
Foundational probability toolkit
$P(A)=\frac{n(A)}{n(U)},\qquad P(A')=1-P(A),\qquad \widehat{P}(A)=\frac{f_A}{N}$
Probability scale and equally likely outcomes
$0\le P(A)\le1,\qquad P(A)=\frac{n(A)}{n(U)}$
Complement, relative frequency and expectation
$P(A')=1-P(A),\qquad \widehat{P}(A)=\frac{f_A}{N},\qquad E(\text{count})=NP(A)$
Topic 4.6Not started
Probability diagrams and combined events
Venn diagrams, tree diagrams and tables organize multi-stage or overlapping events. Accurate structure is often the key step before applying formulas.
190 min estimated
Topic 4.6 progress0%
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Learning objectives
Represent outcomes with Venn diagrams, tree diagrams, sample-space diagrams or tables.
Use the addition rule for unions.
Recognize mutually exclusive and independent events.
Calculate conditional probabilities.
Key formulas
Two-set region counts
$n(A\text{ only})=n(A)-n(A\cap B),\qquad n((A\cup B)')=n(U)-n(A\cup B)$
De Morgan’s laws
$(A\cup B)'=A'\cap B',\qquad (A\cap B)'=A'\cup B'$
General addition rule
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Mutually exclusive condition
$A\cap B=\varnothing\qquad\Longleftrightarrow\qquad P(A\cap B)=0$
Disjoint addition
$P(A\cup B)=P(A)+P(B)$
Tree-diagram strategy
$P(E)=\sum\limits_{j=1}^{m}\prod_{k=1}^{s_j}p_{jk}$
Conditional probability
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B)>0$
Conditional probability from counts
$P(A\mid B)=\frac{n(A\cap B)}{n(B)}$
General multiplication rule
$P(A\cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$
Three-stage path
$P(A\cap B\cap C)=P(A)P(B\mid A)P(C\mid A\cap B)$
Independent-events product condition
$P(A\cap B)=P(A)P(B)$
Equivalent conditional condition
$P(A\mid B)=P(A),\qquad P(B)>0$
Combined-events toolkit
$P(A\cup B)=P(A)+P(B)-P(A\cap B),\qquad P(A\mid B)=\frac{P(A\cap B)}{P(B)}$
Addition and complement
$P(A\cup B)=P(A)+P(B)-P(A\cap B),\qquad P(A')=1-P(A)$
Conditional and multiplication rules
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(A\cap B)=P(A)P(B\mid A)$
Independence test
$A\text{ and }B\text{ independent}\quad\Longleftrightarrow\quad P(A\cap B)=P(A)P(B)$
Topic 4.7Not started
Discrete random variables
A discrete random variable assigns a numerical value to each outcome. Students should interpret a probability distribution and use expected value as the long-run mean of the model.
175 min estimated
Topic 4.7 progress0%
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Learning objectives
Construct or interpret a probability distribution.
Check that probabilities are valid and total 1.
Calculate and interpret expected value in context.
Key formulas
Valid probability values
$0\le P(X=x_i)\le1\quad\text{for }i=1,2,\ldots,n$
Total probability
$\sum\limits_{i=1}^{n}P(X=x_i)=1$
Normalization equation
$\sum\limits_{i=1}^{n}p_i=1$
Expected value
$E(X)=\mu=\sum\limits_{i=1}^{n}x_iP(X=x_i)$
Fair-game condition
$E(X)=0\quad\text{when }X\text{ is the player's net gain}$
Second moment
$E(X^2)=\sum\limits_{i=1}^{n}x_i^2P(X=x_i)$
Variance
$\operatorname{Var}(X)=\sum\limits_{i=1}^{n}(x_i-\mu)^2P(X=x_i)=E(X^2)-[E(X)]^2$
Standard deviation
$\sigma_X=\sqrt{\operatorname{Var}(X)}$
Expectation under a linear transformation
$E(aX+b)=aE(X)+b$
Variance under a linear transformation
$\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$
Standard deviation under a linear transformation
$\sigma_{aX+b}=|a|\sigma_X$
Probability total
$\sum\limits_{i=1}^{n}P(X=x_i)=1$
Expected value
$E(X)=\sum\limits_{i=1}^{n}x_iP(X=x_i)$
Variance shortcut
$\operatorname{Var}(X)=E(X^2)-[E(X)]^2$
Fair-game condition
$E(X)=0\quad\text{for signed net gain }X$
Topic 4.8Not started
Binomial distribution
The binomial model applies to a fixed number of independent trials with two outcomes and constant success probability. Students should first verify that those conditions make sense.
180 min estimated
Topic 4.8 progress0%
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Learning objectives
Use X ~ B(n,p).
Calculate exact, cumulative or interval probabilities with technology.
Use the mean and variance of a binomial distribution.
Key formulas
Binomial notation
$X\sim B(n,p)$
Failure probability
$q=1-p$
Binomial support
$x\in\{0,1,2,\ldots,n\}$
Exact binomial probability
$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$
Combination
$\binom{n}{x}=\frac{n!}{x!(n-x)!}$
GDC exact-probability input
$\operatorname{binompdf}(n,p,x)=P(X=x)$
GDC cumulative-probability input
$\operatorname{binomcdf}(n,p,x)=P(X\le x)$
At least r successes
$P(X\ge r)=1-P(X\le r-1)$
More than r successes
$P(X>r)=1-P(X\le r)$
Inclusive interval
$P(a\le X\le b)=P(X\le b)-P(X\le a-1)$
Binomial mean
$E(X)=\mu=np$
Binomial variance
$\operatorname{Var}(X)=\sigma^2=np(1-p)=npq$
Binomial standard deviation
$\sigma=\sqrt{np(1-p)}$
Binomial model
$X\sim B(n,p)$
Exact probability
$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$
At least r
$P(X\ge r)=1-P(X\le r-1)$
Mean
$E(X)=np$
Variance
$\operatorname{Var}(X)=np(1-p)$
Standard deviation
$\sigma=\sqrt{np(1-p)}$
Topic 4.9Not started
Normal distribution
The normal distribution models continuous data with a symmetric bell-shaped curve determined by a mean and standard deviation. Students use technology for probabilities and inverse-normal questions.
210 min estimated
Topic 4.9 progress0%
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Learning objectives
Recognize the role of μ and σ.
Calculate probabilities over intervals.
Use inverse normal calculations to find values associated with given cumulative probabilities.
Key formulas
Total probability
$P(-\infty<X<\infty)=1$
Center of a normal distribution
$\text{mean}=\text{median}=\text{mode}=\mu$
Area on either side of the mean
$P(X<\mu)=P(X>\mu)=\frac{1}{2}$
Normal-distribution notation
$X\sim N(\mu,\sigma^2)$
Expected value
$E(X)=\mu$
Variance
$\operatorname{Var}(X)=\sigma^2$
Standard deviation
$\operatorname{SD}(X)=\sigma=\sqrt{\operatorname{Var}(X)}$
Within one standard deviation
$P(\mu-\sigma<X<\mu+\sigma)\approx0.68$
Within two standard deviations
$P(\mu-2\sigma<X<\mu+2\sigma)\approx0.95$
Within three standard deviations
$P(\mu-3\sigma<X<\mu+3\sigma)\approx0.997$
Complement rule
$P(X>a)=1-P(X\le a)$
Interval from cumulative probabilities
$P(a<X<b)=P(X<b)-P(X\le a)$
Probability at one point
$P(X=a)=0$
Mirror-tail symmetry
$P(X<\mu-d)=P(X>\mu+d)$
Each tail outside a central area
$P(X<L)=P(X>U)=\frac{1-c}{2}$
Interval probability on a GDC
$P(a<X<b)=\operatorname{normalcdf}(a,b,\mu,\sigma)$
Lower-tail probability
$P(X<b)=\operatorname{normalcdf}(-\infty,b,\mu,\sigma)$
Upper-tail probability
$P(X>a)=\operatorname{normalcdf}(a,\infty,\mu,\sigma)$
Quantile definition
$P(X\le x_q)=q$
Inverse-normal command
$x_q=\operatorname{invNorm}(q,\mu,\sigma)$
Convert an upper-tail area
$P(X>x)=p\quad\Longrightarrow\quad P(X\le x)=1-p$
Highest p cutoff
$P(X>k)=p\quad\Longrightarrow\quad k=\operatorname{invNorm}(1-p,\mu,\sigma)$
Central interval tail area
$\text{each tail}=\frac{1-c}{2}$
Central interval boundaries
$L=\operatorname{invNorm}\!\left(\frac{1-c}{2},\mu,\sigma\right),\quad U=\operatorname{invNorm}\!\left(\frac{1+c}{2},\mu,\sigma\right)$
Normal density
$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$
Inflection points
$x=\mu-\sigma\quad\text{and}\quad x=\mu+\sigma$
Expected count
$\text{expected count}=nP(\text{event})$
Normal model
$X\sim N(\mu,\sigma^2)$
Total area
$P(-\infty<X<\infty)=1$
Symmetry
$P(X<\mu-d)=P(X>\mu+d)$
Normal CDF
$P(a<X<b)=\operatorname{normalcdf}(a,b,\mu,\sigma)$
Inverse normal
$P(X\le x_q)=q\quad\Longrightarrow\quad x_q=\operatorname{invNorm}(q,\mu,\sigma)$
Topic 4.10Not started
Regression line of x on y (current syllabus)
The regression line depends on which variable is being predicted. Reversing the prediction direction requires the regression of x on y rather than simply rearranging the equation for y on x.
195 min estimated
Topic 4.10 progress0%
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Learning objectives
Select the correct regression equation for the variable being predicted.
Use technology to obtain the appropriate regression model.
Keep predictions within a reasonable data range whenever possible.
Key formulas
Correlation range
$-1\le r\le1$
Regression of y on x
$\hat y=a+bx$
Vertical residual
$e_y=y-\hat y$
Least-squares objective for y on x
$\min\sum\limits_{i=1}^{n}(y_i-\hat y_i)^2$
Regression of x on y
$\hat x=c+dy$
Horizontal residual
$e_x=x-\hat x$
Least-squares objective for x on y
$\min\sum\limits_{i=1}^{n}(x_i-\hat x_i)^2$
Centroid property
$(\bar x,\bar y)\text{ lies on both regression lines}$
Slope of y on x
$b_{yx}=r\frac{s_y}{s_x}$
Slope of x on y
$b_{xy}=r\frac{s_x}{s_y}$
Product of regression slopes
$b_{yx}b_{xy}=r^2$
Slope units
$\text{slope units}=\frac{\text{response units}}{\text{explanatory units}}$
Rearrangement that is not a new regression
$\hat x=c+dy\quad\Longrightarrow\quad y=\frac{\hat x-c}{d}$
Exact-inverse special case
$|r|=1\quad\Longrightarrow\quad b_{yx}b_{xy}=1$
Predict y from x
$\hat y=a+bx$
Predict x from y
$\hat x=c+dy$
Residual directions
$e_y=y-\hat y,\qquad e_x=x-\hat x$
Centroid
$(\bar x,\bar y)\text{ lies on both lines}$
Slope check
$b_{yx}b_{xy}=r^2$
Topic 4.11Not started
Formal conditional probability and independence
Conditional probability updates the sample space after information is known. Independence means that knowing one event occurred does not change the probability of the other.
180 min estimated
Topic 4.11 progress0%
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Learning objectives
Use the formal conditional-probability formula.
Test independence using product or conditional-probability relationships.
Distinguish independence from mutually exclusive events.
Key formulas
Complement rule
$P(A')=1-P(A)$
General addition rule
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Conditional probability
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\quad P(B)>0$
Conditional probability from counts
$P(A\mid B)=\frac{n(A\cap B)}{n(B)}$
Multiplication rule
$P(A\cap B)=P(B)P(A\mid B)$
Equivalent order
$P(A\cap B)=P(A)P(B\mid A)$
Three-event chain rule
$P(A\cap B\cap C)=P(A)P(B\mid A)P(C\mid A\cap B)$
Complementary branches
$P(A\mid B)+P(A'\mid B)=1$
Conditional independence test
$P(A\mid B)=P(A)$
Reverse conditional test
$P(B\mid A)=P(B)$
Product test for independence
$P(A\cap B)=P(A)P(B)$
IB independence statement
$P(A\mid B)=P(A)=P(A\mid B')$
Complement product
$P(A\cap B')=P(A)P(B')$
Mutual exclusivity
$P(A\cap B)=0$
General multiplication rule
$P(A\cap B)=P(B)P(A\mid B)$
Independent product rule
$P(A\cap B)=P(A)P(B)$
Independent conditional rule
$P(A\mid B)=P(A)=P(A\mid B')$
Mutually exclusive rule
$P(A\cap B)=0$
Topic 4.12Not started
Standardization of normal variables
Standardization measures how many standard deviations a value lies above or below the mean. It allows results from different normal distributions to be compared on the same scale.
180 min estimated
Topic 4.12 progress0%
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Learning objectives
Convert a raw value to a z-score.
Use standardized values together with normal probability calculations.
Use inverse methods when parameters or cutoffs must be inferred.
Key formulas
Standard normal notation
$Z\sim N(0,1)$
Standard normal center
$E(Z)=0,\qquad \operatorname{SD}(Z)=1$
Standardization formula
$z=\frac{x-\mu}{\sigma},\qquad \sigma>0$
Random-variable transformation
$Z=\frac{X-\mu}{\sigma}$
Inverse standardization
$x=\mu+z\sigma$
Equal-distance symmetry
$P(Z<-a)=P(Z>a),\qquad a>0$
Standardized distribution
$X\sim N(\mu,\sigma^2)\quad\Longrightarrow\quad Z=\frac{X-\mu}{\sigma}\sim N(0,1)$
Preserved cumulative probability
$P(X\le x)=P\!\left(Z\le\frac{x-\mu}{\sigma}\right)$
Raw interval to standard interval
$P(a<X<b)=P\!\left(\frac{a-\mu}{\sigma}<Z<\frac{b-\mu}{\sigma}\right)$
Right-tail complement
$P(Z>z)=1-P(Z\le z)$
Separate standardizations
$z_A=\frac{x_A-\mu_A}{\sigma_A},\qquad z_B=\frac{x_B-\mu_B}{\sigma_B}$
Percentile conversion
$x_p=\mu+z_p\sigma$
Mean from a standardized cutoff
$\mu=x-z\sigma$
Standard deviation from a cutoff
$\sigma=\frac{x-\mu}{z},\qquad z\ne0$
Raw value to z-score
$z=\frac{x-\mu}{\sigma}$
z-score to raw value
$x=\mu+z\sigma$
Standard normal transformation
$X\sim N(\mu,\sigma^2)\Longrightarrow Z=\frac{X-\mu}{\sigma}\sim N(0,1)$
Matching cumulative areas
$P(X\le x)=P\!\left(Z\le\frac{x-\mu}{\sigma}\right)$
Raw interval conversion
$P(a<X<b)=P\!\left(\frac{a-\mu}{\sigma}<Z<\frac{b-\mu}{\sigma}\right)$
Topic 4.13Not started
Bayes’ theorem
Bayes’ theorem reverses a conditional probability and updates a prior belief after evidence is observed. This topic develops the formula from conditional probability, organizes two- and multi-case models with trees and tables, and applies Bayesian reasoning to IB HL problems.
210 min estimated
Topic 4.13 progress0%
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Learning objectives
Interpret prior, likelihood, evidence and posterior probabilities in context.
Derive and use Bayes’ theorem from the conditional probability definition.
Use the law of total probability for two-case and multi-case partitions.
Find reverse conditional probabilities using formulas, trees and probability tables.
Solve multi-stage Bayesian problems while stating any conditional-independence assumptions.
Use natural frequencies and reasonableness checks to interpret rare-event results.
Apply calculator and GDC methods without losing exact working or mathematical justification.
Communicate an IB HL solution with defined events, structured working and a contextual conclusion.
Key formulas
Conditional probability
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B)>0$
General multiplication rule
$P(A\cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$
Bayesian update structure
$\text{posterior}=\frac{\text{likelihood}\times\text{prior}}{\text{evidence}}$
Bayes’ theorem
$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)},\qquad P(B)>0$
Law of total probability
$P(E)=\sum\limits_{i=1}^{n}P(E\mid H_i)P(H_i)$
Two-case Bayes formula
$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B\mid A)P(A)+P(B\mid A^c)P(A^c)}$
Tree-path multiplication
$P(H_i\cap E)=P(H_i)P(E\mid H_i)$
Evidence from tree leaves
$P(E)=\sum\limits_{i=1}^{n}P(H_i\cap E)$
Conditional probability from counts
$P(A\mid B)=\frac{n(A\cap B)}{n(B)}$
General Bayes formula for a partition
$P(H_j\mid E)=\frac{P(E\mid H_j)P(H_j)}{\sum\limits_{i=1}^{n}P(E\mid H_i)P(H_i)}$
Conditionally independent evidence
$P(E_1\cap E_2\mid H_i)=P(E_1\mid H_i)P(E_2\mid H_i)$
Conditional probability
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$
Bayes’ theorem
$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$
Two-case total probability
$P(B)=P(B\mid A)P(A)+P(B\mid A^c)P(A^c)$
General total probability
$P(E)=\sum\limits_{i=1}^{n}P(E\mid H_i)P(H_i)$
General Bayes formula
$P(H_j\mid E)=\frac{P(E\mid H_j)P(H_j)}{\sum\limits_{i=1}^{n}P(E\mid H_i)P(H_i)}$
Topic 4.14Not started
Variance and continuous random variables
Develop variance from expected values, model continuous random variables with probability density and cumulative distribution functions, and calculate probabilities and moments using integration.
220 min estimated
Topic 4.14 progress0%
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Learning objectives
Calculate expected values, second moments, variance and standard deviation for discrete random variables.
Apply linear transformations to the mean, variance and standard deviation of a random variable.
Explain the meaning of a continuous random variable and distinguish density from probability.
Verify that a function is a valid probability density function and determine unknown constants by normalization.
Use definite integration to find interval probabilities and construct cumulative distribution functions.
Calculate the expected value, second moment, variance and standard deviation of a continuous random variable.
Use a GDC efficiently while retaining the mathematical setup required for IB method marks.
Solve multi-step IB HL problems involving transformations, quantiles and continuous distributions.
Key formulas
Discrete expected value
$E(X)=\sum\limits_{i=1}^{n}x_i p_i$
Variance definition
$\operatorname{Var}(X)=E\!\left[(X-\mu)^2\right],\qquad \mu=E(X)$
Standard deviation
$\sigma_X=\sqrt{\operatorname{Var}(X)}$
Discrete second moment
$E(X^2)=\sum\limits_{i=1}^{n}x_i^2p_i$
Variance identity
$\operatorname{Var}(X)=E(X^2)-[E(X)]^2$
Transformed mean
$E(aX+b)=aE(X)+b$
Transformed variance
$\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$
Transformed standard deviation
$\sigma_{aX+b}=|a|\sigma_X$
Linearity of expectation
$E(aX+bY+c)=aE(X)+bE(Y)+c$
Variance of a linear combination
$\operatorname{Var}(aX+bY)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y)+2ab\operatorname{Cov}(X,Y)$
Independent case
$\operatorname{Var}(aX+bY)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y)$
Point probability
$P(X=a)=0$
Interval probability
$P(a\le X\le b)=\int_a^b f(x)\,dx$
PDF conditions
$f(x)\ge0,\qquad \int_{-\infty}^{\infty}f(x)\,dx=1$
Left-tail probability
$P(X\le b)=\int_{-\infty}^{b}f(x)\,dx$
Right-tail probability
$P(X>b)=1-P(X\le b)$
CDF from a density
$F(x)=P(X\le x)=\int_{-\infty}^{x}f(t)\,dt$
Density from a CDF
$f(x)=F'(x)$
Probability from a CDF
$P(a<X\le b)=F(b)-F(a)$
Quantile equation
$F(q_p)=p,\qquad 0<p<1$
Continuous expected value
$E(X)=\int_{-\infty}^{\infty}x f(x)\,dx$
Expectation of a function
$E[g(X)]=\int_{-\infty}^{\infty}g(x)f(x)\,dx$
Continuous second moment
$E(X^2)=\int_{-\infty}^{\infty}x^2f(x)\,dx$
Continuous variance
$\operatorname{Var}(X)=\int_{-\infty}^{\infty}x^2f(x)\,dx-[E(X)]^2$
Normalization equation
$\int_{\text{support}} f(x)\,dx=1$
Discrete mean
$E(X)=\sum\limits_{i=1}^{n}x_i p_i$
Variance
$\operatorname{Var}(X)=E(X^2)-[E(X)]^2$
Linear transformation
$E(aX+b)=aE(X)+b,\quad \operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$
PDF validity
$f(x)\ge0,\quad \int_{-\infty}^{\infty}f(x)\,dx=1$
Interval probability
$P(a\le X\le b)=\int_a^bf(x)\,dx$
CDF
$F(x)=\int_{-\infty}^{x}f(t)\,dt,\quad f(x)=F'(x)$
Continuous moments
$E(X)=\int xf(x)\,dx,\quad E(X^2)=\int x^2f(x)\,dx$
Unit 5
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Calculus
Calculus is the largest analytical component of AA HL. The SL core establishes differentiation, integration, optimization, motion and area; Higher Level develops rigorous limits, first-principles differentiation, l’Hôpital’s rule, implicit differentiation, inverse-trigonometric and reciprocal-trigonometric calculus, advanced integration, volumes, differential equations and Maclaurin series.
19 topics
Unit 5 progress0%
Topics in this unit
Topic 5.1Not started
Limits, gradients and rates of change
Calculus begins with the idea of what a function approaches and how quickly a quantity changes. Students connect graphical slope, average rate and instantaneous rate.
240 min estimated
Topic 5.1 progress0%
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Learning objectives
Interpret simple limit notation.
Understand the derivative as the gradient function.
Interpret derivatives as rates of change in context.
Key formulas
Finite limit
$\lim_{x\to a}f(x)=L$
Continuity at a point
$\lim_{x\to a}f(x)=f(a)$
Direct-substitution rule for a continuous function
$\lim_{x\to a}f(x)=f(a)$
Difference-of-squares limit
$\lim_{x\to a}\frac{x^2-a^2}{x-a}=2a$
Condition for a two-sided limit
$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\quad\Longrightarrow\quad\lim_{x\to a}f(x)=L$
Average rate of change
$\frac{f(b)-f(a)}{b-a},\qquad a\ne b$
Secant gradient using an increment
$\frac{f(x+h)-f(x)}{h},\qquad h\ne0$
Instantaneous rate at x
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Difference quotient
$\frac{f(x+h)-f(x)}{h},\qquad h\ne0$
Derivative from first principles
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Derivative at a specific point
$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$
Equivalent derivative notation
$f'(x)=y'=\frac{dy}{dx}$
Tangent equation at x=a
$y-f(a)=f'(a)(x-a)$
Rate units
$\text{derivative units}=\frac{\text{output units}}{\text{input units}}$
Velocity from displacement
$v(t)=\frac{ds}{dt}$
Centered-difference estimate
$f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}$
Average rate
$\frac{f(b)-f(a)}{b-a}$
Difference quotient
$\frac{f(x+h)-f(x)}{h}$
Derivative
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Tangent line
$y-f(a)=f'(a)(x-a)$
Topic 5.2Not started
Increasing and decreasing functions
The sign of the first derivative describes local direction of change. This is a central connection between a function and its derivative graph.
220 min estimated
Topic 5.2 progress0%
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Learning objectives
Use $f'(x) > 0$ for increasing intervals.
Use $f'(x) < 0$ for decreasing intervals.
Recognize stationary points where $f'(x) = 0$ , while checking the surrounding behavior.
Key formulas
Increasing test
$f'(x)>0$
Decreasing test
$f'(x)<0$
Stationary-value condition
$f'(c)=0$
Critical-value search
$f'(x)=0\quad\text{or}\quad f'(x)\text{ is undefined}$
Local maximum sign change
$f':\ +\longrightarrow-$
Local minimum sign change
$f':\ -\longrightarrow+$
No extremum
$f':\ +\longrightarrow+\quad\text{or}\quad f':\ -\longrightarrow-$
Increasing
$f'(x)>0$
Decreasing
$f'(x)<0$
Stationary value
$f'(c)=0$
Local maximum
$f':\ +\longrightarrow-$
Local minimum
$f':\ -\longrightarrow+$
Topic 5.3Not started
Derivatives of polynomials
Students apply the power rule term by term and simplify derivatives efficiently. Polynomial differentiation becomes a foundation for tangents, optimization and kinematics.
220 min estimated
Topic 5.3 progress0%
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Learning objectives
Differentiate powers of x and sums of polynomial terms.
Evaluate a derivative at a specified x-value.
Use the derivative to solve a gradient or rate problem.
Key formulas
Derivative from first principles
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Equivalent derivative notation
$f'(x)=y'=\frac{dy}{dx}$
Gradient at x=a
$m_{\text{tangent}}=f'(a)$
Power rule
$\frac{d}{dx}\left(x^n\right)=nx^{n-1}$
Power rule with a constant coefficient
$\frac{d}{dx}\left(ax^n\right)=anx^{n-1}$
Derivative of a constant
$\frac{d}{dx}(c)=0$
Constant multiple rule
$\frac{d}{dx}\left[af(x)\right]=af'(x)$
Derivative of a linear term
$\frac{d}{dx}(ax)=a$
Sum rule
$\frac{d}{dx}\left[f(x)+g(x)\right]=f'(x)+g'(x)$
Difference rule
$\frac{d}{dx}\left[f(x)-g(x)\right]=f'(x)-g'(x)$
General polynomial derivative
$\frac{d}{dx}\left(\sum\limits_{k=0}^{n}a_kx^k\right)=\sum\limits_{k=1}^{n}ka_kx^{k-1}$
Gradient at a point
$m=f'(a)$
Inputs with gradient m
$f'(x)=m$
Tangent equation
$y-f(a)=f'(a)(x-a)$
Normal gradient
$m_{\text{normal}}=-\frac{1}{f'(a)}$
Stationary condition
$f'(c)=0$
Stationary point coordinates
$\left(c,f(c)\right)$
Velocity from displacement
$v(t)=s'(t)$
Acceleration from displacement
$a(t)=v'(t)=s''(t)$
Power rule
$\frac{d}{dx}\left(ax^n\right)=anx^{n-1}$
Constant rule
$\frac{d}{dx}(c)=0$
Gradient at x=a
$m=f'(a)$
Topic 5.4Not started
Tangents and normals
The derivative gives the gradient of a tangent. A normal line is perpendicular to the tangent, so its gradient is the negative reciprocal when both lines have finite non-zero gradient.
220 min estimated
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Learning objectives
Find the tangent gradient from f′(a).
Write the equation of the tangent through the point on the curve.
Find a normal gradient and equation.
Key formulas
Tangent gradient
$m_{\text{tangent}}=f'(a)$
Point-slope form
$y-y_1=m(x-x_1)$
Tangent at x=a
$y-f(a)=f'(a)(x-a)$
Perpendicular-gradient condition
$m_1m_2=-1$
Normal gradient
$m_{\text{normal}}=-\frac{1}{f'(a)}$
Normal at x=a
$y-f(a)=-\frac{1}{f'(a)}(x-a)$
Given tangent gradient
$f'(x)=m$
Given normal gradient
$f'(x)=-\frac{1}{n}$
Horizontal tangent
$f'(a)=0\quad\Longrightarrow\quad y=f(a)$
Corresponding vertical normal
$x=a$
Parallel-gradient condition
$m_1=m_2$
Tangent approximation
$f(x)\approx f(a)+f'(a)(x-a)$
Tangent equation
$y-f(a)=f'(a)(x-a)$
Normal equation
$y-f(a)=-\frac{1}{f'(a)}(x-a)$
Perpendicular gradients
$m_{\text{tangent}}m_{\text{normal}}=-1$
Topic 5.5Not started
Introduction to integration
Integration is introduced as reverse differentiation and as accumulation/area. Students should understand the role of the constant of integration for antiderivatives and the meaning of a definite integral.
220 min estimated
Topic 5.5 progress0%
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Learning objectives
Find simple antiderivatives.
Use a boundary or initial condition to determine the constant C.
Use technology to evaluate definite integrals and interpret area when the function is non-negative.
Key formulas
Antiderivative condition
$F'(x)=f(x)$
Reverse relationship
$\frac{d}{dx}\left(\int f(x)\,dx\right)=f(x)$
Indefinite integral notation
$\int f(x)\,dx=F(x)+C\quad\text{when }F'(x)=f(x)$
Definite integral notation
$\int_a^b f(x)\,dx$
Integration power rule
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\neq-1$
Constant multiple rule
$\int kf(x)\,dx=k\int f(x)\,dx$
Sum and difference rule
$\int\bigl(f(x)\pm g(x)\bigr)\,dx=\int f(x)\,dx\pm\int g(x)\,dx$
Integral of a constant
$\int k\,dx=kx+C$
Family of antiderivatives
$F(x)+C,\qquad C\in\mathbb{R}$
Polynomial rule
$\int\left(\sum\limits_{k=0}^{m}a_kx^k\right)dx=\sum\limits_{k=0}^{m}\frac{a_k}{k+1}x^{k+1}+C$
Condition from a point
$F(a)=b\quad\text{for a point }(a,b)$
Verification statement
$\frac{d}{dx}\bigl(F(x)+C\bigr)=f(x)$
Initial-value pattern
$y'=f(x),\ y(a)=b\quad\Longrightarrow\quad y=\int f(x)\,dx\text{ and then use }y(a)=b$
Fundamental Theorem of Calculus
$\int_a^b f(x)\,dx=\bigl[F(x)\bigr]_a^b=F(b)-F(a)$
Motion hierarchy
$a(t)=v'(t),\qquad v(t)=s'(t)$
Displacement from velocity
$s(t)=\int v(t)\,dt$
Antiderivative definition
$F'(x)=f(x)\quad\Longleftrightarrow\quad\int f(x)\,dx=F(x)+C$
Power rule
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\neq-1$
Constant rule
$\int k\,dx=kx+C$
Linearity
$\int(af(x)+bg(x))\,dx=a\int f(x)\,dx+b\int g(x)\,dx$
Definite integral
$\int_a^b f(x)\,dx=F(b)-F(a)$
Topic 5.6Not started
Derivatives of elementary and composite functions
AA HL extends differentiation beyond polynomials to trigonometric, exponential and logarithmic functions. Correct use of the chain, product and quotient rules is essential.
260 min estimated
Topic 5.6 progress0%
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Learning objectives
Differentiate sin x, cos x, $e^{x}$ and ln x.
Apply constant multiple and sum rules.
Use the chain rule for composite functions.
Use product and quotient rules accurately.
Key formulas
Power rule
$\frac{d}{dx}(x^n)=nx^{n-1}$
Constant multiple power
$\frac{d}{dx}(kx^n)=knx^{n-1}$
Natural exponential derivative
$\frac{d}{dx}(e^x)=e^x$
General exponential derivative
$\frac{d}{dx}(a^x)=a^x\ln a,\qquad a>0,\ a\neq1$
Natural logarithm derivative
$\frac{d}{dx}(\ln x)=\frac{1}{x},\qquad x>0$
General logarithm derivative
$\frac{d}{dx}(\log_a x)=\frac{1}{x\ln a},\qquad x>0$
Derivative of sine
$\frac{d}{dx}(\sin x)=\cos x$
Derivative of cosine
$\frac{d}{dx}(\cos x)=-\sin x$
Derivative of tangent
$\frac{d}{dx}(\tan x)=\sec^2x$
Composite notation
$y=f(g(x))=(f\circ g)(x)$
Chain rule
$\frac{d}{dx}\bigl(f(g(x))\bigr)=f'(g(x))g'(x)$
Leibniz chain notation
$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$
Composite power rule
$\frac{d}{dx}\bigl([g(x)]^n\bigr)=n[g(x)]^{n-1}g'(x)$
Exponential chain rule
$\frac{d}{dx}\bigl(e^{g(x)}\bigr)=e^{g(x)}g'(x)$
General exponential chain rule
$\frac{d}{dx}\bigl(a^{g(x)}\bigr)=a^{g(x)}\ln(a)\,g'(x)$
Logarithmic chain rule
$\frac{d}{dx}\bigl(\ln(g(x))\bigr)=\frac{g'(x)}{g(x)}$
Sine chain rule
$\frac{d}{dx}\bigl(\sin(g(x))\bigr)=\cos(g(x))g'(x)$
Cosine chain rule
$\frac{d}{dx}\bigl(\cos(g(x))\bigr)=-\sin(g(x))g'(x)$
Tangent chain rule
$\frac{d}{dx}\bigl(\tan(g(x))\bigr)=\sec^2(g(x))g'(x)$
Linearity of differentiation
$\frac{d}{dx}\bigl(af(x)+bg(x)\bigr)=af'(x)+bg'(x)$
Product rule
$\frac{d}{dx}(uv)=u'v+uv'$
Quotient rule
$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2},\qquad v\neq0$
Tangent gradient
$m=f'(a)$
Stationary-point condition
$f'(x)=0$
Natural exponential
$\frac{d}{dx}(e^x)=e^x$
Natural logarithm
$\frac{d}{dx}(\ln x)=\frac{1}{x}$
Trigonometric derivatives
$(\sin x)'=\cos x,\quad(\cos x)'=-\sin x,\quad(\tan x)'=\sec^2x$
Chain rule
$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$
Product rule
$(uv)'=u'v+uv'$
Quotient rule
$\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$
Topic 5.7Not started
Second derivative
The second derivative measures how the first derivative changes. Graphically it connects to concavity and helps classify stationary behavior.
230 min estimated
Topic 5.7 progress0%
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Learning objectives
Relate f, f′ and f″ through their graphs.
Interpret the sign of f″ as concavity information.
Use f″ as part of local-max/min analysis where appropriate.
Key formulas
Second derivative definition
$f''(x)=\frac{d}{dx}\bigl(f'(x)\bigr)$
Equivalent notation
$f''(x)=y''=\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$
Successive derivative process
$f(x)\xrightarrow{\frac{d}{dx}}f'(x)\xrightarrow{\frac{d}{dx}}f''(x)$
Exponential pattern
$\frac{d^2}{dx^2}\bigl(e^{kx}\bigr)=k^2e^{kx}$
Sine pattern
$\frac{d^2}{dx^2}\bigl(\sin(kx)\bigr)=-k^2\sin(kx)$
Cosine pattern
$\frac{d^2}{dx^2}\bigl(\cos(kx)\bigr)=-k^2\cos(kx)$
Concavity conditions
$f''(x)>0\Rightarrow\text{concave up},\qquad f''(x)<0\Rightarrow\text{concave down}$
Derivative hierarchy
$f\xrightarrow{\text{gradient}}f'\xrightarrow{\text{gradient}}f''$
Example derivatives
$f(x)=x^3-3x,\qquad f'(x)=3x^2-3,\qquad f''(x)=6x$
Second-derivative test
$f'(a)=0,\quad f''(a)>0\Rightarrow\text{local minimum},\quad f''(a)<0\Rightarrow\text{local maximum}$
Inflection requirement
$f''\text{ changes sign at }x=a$
Motion derivative hierarchy
$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t)$
Second derivative
$f''(x)=\frac{d}{dx}\bigl(f'(x)\bigr)=\frac{d^2y}{dx^2}$
Concavity
$f''>0\Rightarrow\text{concave up},\qquad f''<0\Rightarrow\text{concave down}$
Stationary-point test
$f'(a)=0:\quad f''(a)>0\Rightarrow\min,\quad f''(a)<0\Rightarrow\max$
Motion hierarchy
$v=s',\qquad a=v'=s''$
Topic 5.8Not started
Local maxima, minima and points of inflection
Optimization problems combine calculus with modelling. Students must identify the quantity to optimize, find critical points and justify the final choice within the allowed domain.
260 min estimated
Topic 5.8 progress0%
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Learning objectives
Find and classify local extrema.
Solve optimization problems with constraints.
Identify points of inflection, including cases with zero or non-zero gradient.
Interpret stationary points in the original context.
Key formulas
Stationary-point condition
$f'(a)=0$
Local minimum test
$f'(a)=0,\quad f''(a)>0$
Local maximum test
$f'(a)=0,\quad f''(a)<0$
Concave up
$f''(x)>0$
Concave down
$f''(x)<0$
Inflection inputs for the visual
$x=\pm\sqrt{\frac{2}{3}}$
Inflection coordinates
$\left(\pm\sqrt{\frac{2}{3}},-\frac{20}{9}\right)$
Stationary condition
$f'(a)=0$
First-derivative maximum pattern
$f':\quad +\;\to\;-$
First-derivative minimum pattern
$f':\quad -\;\to\;+$
Second-derivative minimum test
$f'(a)=0,\quad f''(a)>0$
Second-derivative maximum test
$f'(a)=0,\quad f''(a)<0$
Inflection requirement
$f''\text{ changes sign at }x=a$
Topic 5.9Not started
Kinematics
One-dimensional motion links displacement, velocity and acceleration through differentiation and integration. Direction matters: distance travelled is not always the same as displacement.
280 min estimated
Topic 5.9 progress0%
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Learning objectives
Differentiate displacement to obtain velocity and velocity to obtain acceleration.
Integrate acceleration or velocity when conditions are given.
Find total distance by accounting for changes of direction.
Key formulas
Velocity from displacement
$v(t)=\frac{ds}{dt}=s'(t)$
Acceleration from velocity
$a(t)=\frac{dv}{dt}=v'(t)=s''(t)$
Average velocity
$\bar v=\frac{s(t_2)-s(t_1)}{t_2-t_1}$
Speed
$\text{speed}=|v(t)|$
Velocity change
$v(t_2)-v(t_1)=\int_{t_1}^{t_2}a(t)\,dt$
Displacement change
$s(t_2)-s(t_1)=\int_{t_1}^{t_2}v(t)\,dt$
Displacement maximum test
$v(r)=0,\quad a(r)<0$
Displacement minimum test
$v(r)=0,\quad a(r)>0$
Net displacement
$\Delta s=s(t_2)-s(t_1)$
Total distance
$D=\int_{t_1}^{t_2}|v(t)|\,dt$
Derivative chain
$s(t)\xrightarrow{\frac{d}{dt}}v(t)\xrightarrow{\frac{d}{dt}}a(t)$
Total distance on an interval
$D=\int_{t_1}^{t_2}|v(t)|\,dt$
Velocity
$v(t)=s'(t)=\frac{ds}{dt}$
Acceleration
$a(t)=v'(t)=s''(t)=\frac{d^2s}{dt^2}$
Topic 5.10Not started
Indefinite integrals and reverse chain rule
Students integrate a wider family of functions and recognize linear-composite patterns. Integration by inspection is often faster than formal substitution when the inner function is linear.
300 min estimated
Topic 5.10 progress0%
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Learning objectives
Integrate rational powers $x^{n}$ where valid.
Integrate $\frac{1}{x}$ and $e^{x}$ .
Integrate simple composite functions of ax+b by reversing the chain rule.
Recognize common trigonometric antiderivatives.
Key formulas
Indefinite integral definition
$\int f(x)\,dx=F(x)+C\quad\text{when }F'(x)=f(x)$
Power rule
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\ne-1$
Constant multiple
$\int kf(x)\,dx=k\int f(x)\,dx$
Sum and difference
$\int(f(x)\pm g(x))\,dx=\int f(x)\,dx\pm\int g(x)\,dx$
Integral of a constant
$\int k\,dx=kx+C$
Reciprocal
$\int\frac{1}{x}\,dx=\ln|x|+C$
Natural exponential
$\int e^x\,dx=e^x+C$
Cosine
$\int\cos x\,dx=\sin x+C$
Sine
$\int\sin x\,dx=-\cos x+C$
Secant squared
$\int\sec^2x\,dx=\tan x+C$
General reverse-chain form
$\int g'(x)f(g(x))\,dx=F(g(x))+C\quad\text{where }F'(u)=f(u)$
Composite power
$\int g'(x)[g(x)]^n\,dx=\frac{[g(x)]^{n+1}}{n+1}+C,\quad n\ne-1$
Linear-composite power
$\int(ax+b)^n\,dx=\frac{(ax+b)^{n+1}}{a(n+1)}+C,\quad a\ne0,\ n\ne-1$
Logarithmic reverse chain
$\int\frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C$
Natural-exponential reverse chain
$\int g'(x)e^{g(x)}\,dx=e^{g(x)}+C$
Linear exponential
$\int e^{ax+b}\,dx=\frac{1}{a}e^{ax+b}+C,\quad a\ne0$
Cosine reverse chain
$\int g'(x)\cos(g(x))\,dx=\sin(g(x))+C$
Sine reverse chain
$\int g'(x)\sin(g(x))\,dx=-\cos(g(x))+C$
Secant-squared reverse chain
$\int g'(x)\sec^2(g(x))\,dx=\tan(g(x))+C$
Verification principle
$\frac{d}{dx}\left(\int f(x)\,dx\right)=f(x)$
Composite power
$\int g'(x)[g(x)]^n\,dx=\frac{[g(x)]^{n+1}}{n+1}+C$
Logarithmic pattern
$\int\frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C$
Exponential pattern
$\int g'(x)e^{g(x)}\,dx=e^{g(x)}+C$
Cosine pattern
$\int g'(x)\cos(g(x))\,dx=\sin(g(x))+C$
Sine pattern
$\int g'(x)\sin(g(x))\,dx=-\cos(g(x))+C$
Topic 5.11Not started
Definite integrals and areas between curves
A definite integral gives signed accumulation. When a question asks for geometric area, negative contributions must be handled carefully and regions may need to be split.
300 min estimated
Topic 5.11 progress0%
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Learning objectives
Evaluate definite integrals analytically when possible.
Find areas between a curve and the x-axis.
Find areas between two curves using upper-minus-lower function.
Find intersections first when they define integration limits.
Key formulas
Signed-area interpretation
$\int_a^b f(x)\,dx=\text{area above the }x\text{-axis}-\text{area below the }x\text{-axis}$
Fundamental theorem of calculus
$\int_a^b f(x)\,dx=[F(x)]_a^b=F(b)-F(a)\quad\text{where }F'(x)=f(x)$
Reverse limits
$\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$
Equal limits
$\int_a^a f(x)\,dx=0$
Split an interval
$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx$
Area above the x-axis
$A=\int_a^b f(x)\,dx\quad\text{when }f(x)\ge0$
Area below the x-axis
$A=-\int_a^b f(x)\,dx=\int_a^b[-f(x)]\,dx\quad\text{when }f(x)\le0$
Total area
$A_{\mathrm{total}}=\int_a^b|f(x)|\,dx$
Intersection condition
$f(x)=g(x)$
Area between two curves
$A=\int_a^b[\text{upper}(x)-\text{lower}(x)]\,dx$
Fundamental theorem
$\int_a^b f(x)\,dx=F(b)-F(a)$
Split interval
$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx$
Area above axis
$A=\int_a^b f(x)\,dx\quad\text{if }f(x)\ge0$
Area below axis
$A=-\int_a^b f(x)\,dx\quad\text{if }f(x)\le0$
Area between curves
$A=\int_a^b[\text{upper}(x)-\text{lower}(x)]\,dx$
Topic 5.12Not started
Continuity, differentiability and the derivative from first principles
Use one-sided limits to test continuity, classify discontinuities, distinguish continuity from differentiability, and construct derivatives directly from the limiting difference quotient.
225 min estimated
Topic 5.12 progress0%
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Learning objectives
Interpret continuity locally, graphically and through limits.
Use left-hand and right-hand limits to decide whether a two-sided limit exists.
Apply all three conditions for continuity at a point and classify common discontinuities.
Determine unknown parameters that make piecewise functions continuous or differentiable.
Interpret a difference quotient as a secant gradient and its limit as a tangent gradient.
Derive derivatives from first principles using algebraic simplification and limits.
Explain why differentiability implies continuity and provide counterexamples to the converse.
Recognize corners, cusps, vertical tangents and discontinuities from graphs.
Use a GDC to investigate and verify while retaining rigorous limit-based reasoning.
Key formulas
Continuity at a point
$\lim_{x\to c}f(x)=f(c)$
Existence of a two-sided limit
$\lim_{x\to c}f(x)=L\iff\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=L$
Three-condition test
$f(c)\text{ exists},\quad \lim_{x\to c}f(x)\text{ exists},\quad \lim_{x\to c}f(x)=f(c)$
Endpoint continuity
$\lim_{x\to a^+}f(x)=f(a),\qquad \lim_{x\to b^-}f(x)=f(b)$
Derivative at a point
$f'(c)=\lim_{h\to0}\frac{f(c+h)-f(c)}{h}$
Difference quotient
$\frac{f(x+h)-f(x)}{h},\qquad h\ne0$
Alternative derivative definition
$f'(c)=\lim_{x\to c}\frac{f(x)-f(c)}{x-c}$
Derivative function from first principles
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Quadratic derivative from first principles
$\frac{d}{dx}(ax^2+bx+c)=2ax+b$
Differentiability implication
$f\text{ differentiable at }c\implies f\text{ continuous at }c$
Differentiability at a join
$\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=f(c),\qquad f'_-(c)=f'_+(c)$
Tangent equation
$y-f(c)=f'(c)(x-c)$
Continuity
$\lim_{x\to c}f(x)=f(c)$
Two-sided limit
$\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=L$
Difference quotient
$\frac{f(x+h)-f(x)}{h}$
Derivative from first principles
$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
Derivative at a point
$f'(c)=\lim_{x\to c}\frac{f(x)-f(c)}{x-c}$
Piecewise differentiability
$f'_-(c)=f'_+(c)$
Tangent
$y-f(c)=f'(c)(x-c)$
Topic 5.13Not started
Evaluation of limits: l’Hôpital’s rule and series methods
Evaluate difficult finite and infinite limits by classifying indeterminate forms, choosing an efficient algebraic, l’Hôpital or Maclaurin-series method, and presenting a rigorous IB HL justification.
220 min estimated
Topic 5.13 progress0%
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Learning objectives
Distinguish indeterminate forms from limits that are already determined or divergent.
State and check the conditions for l’Hôpital’s rule at finite points, one-sided endpoints and infinity.
Apply l’Hôpital’s rule once or repeatedly, rechecking the new form after every differentiation.
Rewrite products, differences and variable powers into quotient or logarithmic forms before applying an appropriate method.
Use standard Maclaurin expansions and order notation to expose leading behaviour and cancellation.
Choose efficiently among direct substitution, algebraic simplification, standard limits, l’Hôpital’s rule and series expansions.
Evaluate exponential, logarithmic and trigonometric limits exactly and verify results numerically with a GDC.
Communicate domain conditions, one-sided behaviour and method validity clearly in IB HL solutions.
Key formulas
Limit notation
$\lim_{x\to a}f(x)=L$
Quotient indeterminate forms
$\frac00,\qquad \frac{\infty}{\infty}$
Other indeterminate forms
$0\cdot\infty,\quad \infty-\infty,\quad 0^0,\quad 1^\infty,\quad \infty^0$
l’Hôpital’s rule
$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}$
Single application workflow
$\frac{f}{g}\xrightarrow[\frac{0}{0}\text{ or }\frac{\infty}{\infty}]{\text{l’Hôpital}}\frac{f'}{g'}$
Repeated l’Hôpital
$\lim\frac{f}{g}=\lim\frac{f'}{g'}=\lim\frac{f''}{g''}$
Sine limit
$\lim_{x\to0}\frac{\sin x}{x}=1$
Cosine difference limit
$\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12$
Tangent limit
$\lim_{x\to0}\frac{\tan x}{x}=1$
Exponential standard limit
$\lim_{x\to0}\frac{e^x-1}{x}=1$
Logarithmic standard limit
$\lim_{x\to0}\frac{\ln(1+x)}{x}=1$
Growth hierarchy
$\ln x\ll x^a\ll e^x\quad(x\to\infty,\ a>0)$
Product-to-quotient transformations
$fg=\frac{f}{\frac{1}{g}}=\frac{g}{\frac{1}{f}}$
Logarithmic transformation
$y=f(x)^{g(x)}\quad\Longrightarrow\quad \ln y=g(x)\ln f(x)$
Exponential recovery
$\lim\ln y=L\quad\Longrightarrow\quad\lim y=e^L$
Maclaurin series
$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+\cdots$
Sine local approximation
$\sin x=x-\frac{x^3}{6}+O(x^5)$
Exponential series
$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+O(x^4)$
Sine series
$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+O(x^7)$
Cosine series
$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+O(x^6)$
Logarithm series
$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+O(x^4)$
Geometric series
$\frac1{1-x}=1+x+x^2+x^3+\cdots\quad(|x|<1)$
General binomial series
$(1+x)^p=1+px+\frac{p(p-1)}{2!}x^2+\frac{p(p-1)(p-2)}{3!}x^3+\cdots$
Tangent series
$\tan x=x+\frac{x^3}{3}+O(x^5)$
Leading-term principle
$f(x)=a_mx^m+O(x^{m+1})\quad\Longrightarrow\quad\lim_{x\to0}\frac{f(x)}{x^m}=a_m$
Sine cancellation limit
$\lim_{x\to0}\frac{\sin x-x}{x^3}=-\frac16$
Tangent cancellation limit
$\lim_{x\to0}\frac{\tan x-x}{x^3}=\frac13$
Exponential cancellation limit
$\lim_{x\to0}\frac{e^x-1-x}{x^2}=\frac12$
Logarithmic cancellation limit
$\lim_{x\to0}\frac{\ln(1+x)-x}{x^2}=-\frac12$
Conjugate identity
$(\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})=u-v$
Method-selection chain
$\text{substitute}\to\text{simplify}\to\text{classify}\to\text{choose method}\to\text{validate}$
l’Hôpital checklist
$\frac00\text{ or }\frac{\infty}{\infty}\quad\Longrightarrow\quad\lim\frac{f}{g}=\lim\frac{f'}{g'}$
Core local equivalents
$\sin x\sim x,\quad \tan x\sim x,\quad e^x-1\sim x,\quad \ln(1+x)\sim x$
Core Maclaurin terms
$e^x=1+x+\frac{x^2}{2}+O(x^3),\quad \sin x=x-\frac{x^3}{6}+O(x^5),\quad \cos x=1-\frac{x^2}{2}+O(x^4)$
Topic 5.14Not started
Implicit differentiation, related rates and optimization
Differentiate relationships defined implicitly, use their derivatives for tangents and motion, connect simultaneous rates through a shared constraint, and solve constrained maximum and minimum problems rigorously.
230 min estimated
Topic 5.14 progress0%
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Learning objectives
Distinguish explicit and implicit descriptions of a relationship and recognize when an implicit form is useful.
Apply the chain and product rules while differentiating equations containing both x and y.
Solve an implicit derivative equation for dy/dx and evaluate first or second derivatives at a point.
Find tangent and normal equations, including exceptional horizontal and vertical cases.
Translate verbal related-rate information into variables, a constraint equation and consistently signed derivatives.
Use first- and second-derivative tests, endpoints and domain constraints to justify local or global extrema.
Model geometric and real-world related-rate and optimization situations with correct units and interpretations.
Use a GDC to investigate and verify without replacing analytic differentiation and justification.
Key formulas
Implicit relation
$F(x,y)=0$
Implicit power chain rule
$\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}$
Implicit function chain rule
$\frac{d}{dx}g(y)=g'(y)\frac{dy}{dx}$
Product rule for xy
$\frac{d}{dx}(xy)=x\frac{dy}{dx}+y$
Implicit derivative from F
$\frac{dy}{dx}=-\frac{F_x}{F_y}$
Second implicit derivative
$\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$
Tangent equation
$y-y_0=m_t(x-x_0)$
Normal gradient
$m_n=-\frac1{m_t}$
Time chain rule
$\frac{d}{dt}f(x)=f'(x)\frac{dx}{dt}$
Related-rates product rule
$\frac{d}{dt}(xy)=x\frac{dy}{dt}+y\frac{dx}{dt}$
Sphere volume
$V=\frac43\pi r^3$
Sphere volume rate
$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$
Cone volume rate
$\frac{dV}{dt}=\frac{\pi}{3}\left(2rh\frac{dr}{dt}+r^2\frac{dh}{dt}\right)$
Fixed-length ladder rate
$x\frac{dx}{dt}+y\frac{dy}{dt}=0$
Stationary-point condition
$Q'(x)=0$
Second-derivative maximum test
$Q'(c)=0,\ Q''(c)<0\quad\Longrightarrow\quad\text{local maximum at }c$
Second-derivative minimum test
$Q'(c)=0,\ Q''(c)>0\quad\Longrightarrow\quad\text{local minimum at }c$
Open-box volume
$V(x)=x(L-2x)(W-2x)$
Closed-cylinder surface area
$S=2\pi r^2+2\pi rh$
Fixed-volume cylinder condition
$r^3=\frac{V}{2\pi},\qquad h=2r$
Motion along a curve
$\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}$
Implicit derivative template
$F(x,y)=0\quad\Longrightarrow\quad\frac{dy}{dx}=-\frac{F_x}{F_y}$
Related-rates template
$F(x,y)=0\quad\Longrightarrow\quad F_x\frac{dx}{dt}+F_y\frac{dy}{dt}=0$
Optimization conditions
$Q'(c)=0,\qquad Q''(c)<0\text{ for a local maximum},\qquad Q''(c)>0\text{ for a local minimum}$
Topic 5.15Not started
Derivatives and antiderivatives of special functions
Differentiate and integrate exponential, logarithmic, trigonometric and inverse-trigonometric functions, recognize their composite standard forms, and preserve exact values, domains and constants.
225 min estimated
Topic 5.15 progress0%
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Learning objectives
Differentiate natural and general exponential and logarithmic functions with correct domain conditions.
Differentiate all required trigonometric and inverse-trigonometric functions in radians.
Apply chain, product and quotient rules to combinations of special and algebraic functions.
Recall and verify the corresponding standard antiderivatives, including absolute values and the constant of integration.
Recognize reverse-chain forms involving f prime over f, exponentials, trigonometric functions and inverse-trigonometric patterns.
Adjust for missing constant factors and distinguish standard substitutions from non-standard products.
Evaluate exact derivatives and definite integrals involving pi, logarithms and inverse-trigonometric values.
Use a GDC to check numerical behaviour while presenting exact analytic working in IB HL solutions.
Key formulas
Derivative notation
$\frac{d}{dx}f(x)=f'(x)$
Indefinite integral notation
$\int f(x)\,dx=F(x)+C\quad\text{when }F'(x)=f(x)$
Natural exponential derivative
$\frac{d}{dx}e^x=e^x$
Composite natural exponential
$\frac{d}{dx}e^{u}=u'e^u$
General exponential derivative
$\frac{d}{dx}a^u=u'a^u\ln a$
Natural logarithm derivative
$\frac{d}{dx}\ln x=\frac1x$
Composite logarithm derivative
$\frac{d}{dx}\ln|u|=\frac{u'}u$
General logarithm derivative
$\frac{d}{dx}\log_a u=\frac{u'}{u\ln a}$
Sine derivative
$\frac{d}{dx}\sin u=u'\cos u$
Cosine derivative
$\frac{d}{dx}\cos u=-u'\sin u$
Tangent derivative
$\frac{d}{dx}\tan u=u'\sec^2u$
Secant derivative
$\frac{d}{dx}\sec u=u'\sec u\tan u$
Cosecant derivative
$\frac{d}{dx}\csc u=-u'\csc u\cot u$
Cotangent derivative
$\frac{d}{dx}\cot u=-u'\csc^2u$
Arcsine derivative
$\frac{d}{dx}\arcsin u=\frac{u'}{\sqrt{1-u^2}}$
Arccosine derivative
$\frac{d}{dx}\arccos u=-\frac{u'}{\sqrt{1-u^2}}$
Arctangent derivative
$\frac{d}{dx}\arctan u=\frac{u'}{1+u^2}$
General chain rule
$\frac{d}{dx}F(u(x))=F'(u(x))u'(x)$
Product rule
$(uv)'=u'v+uv'$
Quotient rule
$\left(\frac uv\right)'=\frac{u'v-uv'}{v^2}$
Antiderivative family
$\int f(x)\,dx=F(x)+C\quad\text{if }F'(x)=f(x)$
Natural exponential antiderivative
$\int e^{kx}\,dx=\frac1k e^{kx}+C\quad(k\ne0)$
General exponential antiderivative
$\int a^{kx}\,dx=\frac{a^{kx}}{k\ln a}+C$
Reciprocal antiderivative
$\int\frac1x\,dx=\ln|x|+C$
Logarithm antiderivative
$\int\ln x\,dx=x\ln x-x+C$
Sine antiderivative
$\int\sin x\,dx=-\cos x+C$
Cosine antiderivative
$\int\cos x\,dx=\sin x+C$
Secant-squared antiderivative
$\int\sec^2x\,dx=\tan x+C$
Cosecant-squared antiderivative
$\int\csc^2x\,dx=-\cot x+C$
Secant-tangent antiderivative
$\int\sec x\tan x\,dx=\sec x+C$
Cosecant-cotangent antiderivative
$\int\csc x\cot x\,dx=-\csc x+C$
Tangent antiderivative
$\int\tan x\,dx=-\ln|\cos x|+C$
Cotangent antiderivative
$\int\cot x\,dx=\ln|\sin x|+C$
Secant antiderivative
$\int\sec x\,dx=\ln|\sec x+\tan x|+C$
Cosecant antiderivative
$\int\csc x\,dx=\ln|\csc x-\cot x|+C$
Arcsine antiderivative
$\int\frac{u'}{\sqrt{1-u^2}}\,dx=\arcsin u+C$
Arctangent antiderivative
$\int\frac{u'}{1+u^2}\,dx=\arctan u+C$
Scaled arctangent form
$\int\frac{dx}{a^2+x^2}=\frac1a\arctan\left(\frac xa\right)+C$
Scaled arcsine form
$\int\frac{dx}{\sqrt{a^2-x^2}}=\arcsin\left(\frac xa\right)+C$
Exponential reverse chain
$\int f'e^f\,dx=e^f+C$
Logarithmic reverse chain
$\int\frac{f'}f\,dx=\ln|f|+C$
Cosine reverse chain
$\int f'\cos f\,dx=\sin f+C$
Tangent reverse chain
$\int f'\sec^2f\,dx=\tan f+C$
Exponential–log pair
$\frac{d}{dx}e^u=u'e^u,\qquad \int u'e^u\,dx=e^u+C,\qquad \frac{d}{dx}\ln|u|=\frac{u'}u,\qquad \int\frac{u'}u\,dx=\ln|u|+C$
Core trig pairs
$(\sin u)'=u'\cos u,\quad(\cos u)'=-u'\sin u,\quad(\tan u)'=u'\sec^2u$
Core inverse-trig pairs
$(\arcsin u)'=\frac{u'}{\sqrt{1-u^2}},\qquad(\arctan u)'=\frac{u'}{1+u^2}$
Topic 5.16Not started
Integration by substitution and by parts
Transform composite integrals by substitution, including changed bounds, and reverse the product rule through integration by parts, choosing methods strategically for algebraic and special-function products.
240 min estimated
Topic 5.16 progress0%
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Learning objectives
Review antiderivatives, definite integrals and verification by differentiation.
Choose and execute substitutions that expose standard reverse-chain forms.
Transform the differential and every part of the integrand into one variable.
Change bounds correctly when applying substitution to definite integrals.
Derive and apply integration by parts from the product rule.
Choose u and dv so that differentiation simplifies u and integration of dv is possible.
Use repeated or cyclic integration by parts for polynomial, exponential and trigonometric combinations.
Select efficiently among direct recognition, substitution, algebraic simplification and integration by parts.
Evaluate exact indefinite and definite integrals with correct constants, domains and IB HL justification.
Key formulas
Indefinite integral
$\int f(x)\,dx=F(x)+C\quad\text{when }F'(x)=f(x)$
Fundamental theorem evaluation
$\int_a^b f(x)\,dx=F(b)-F(a)$
Substitution rule
$u=g(x),\quad du=g'(x)\,dx,\quad \int f(g(x))g'(x)\,dx=\int f(u)\,du$
Reverse-chain signal
$g'(x)F(g(x))\quad\longrightarrow\quad u=g(x)$
Exponential substitution pattern
$\int g'e^g\,dx=e^g+C$
Logarithmic substitution pattern
$\int\frac{g'}g\,dx=\ln|g|+C$
Definite substitution
$\int_a^b f(g(x))g'(x)\,dx=\int_{g(a)}^{g(b)}f(u)\,du$
Arctangent reverse chain
$\int\frac{g'}{1+g^2}\,dx=\arctan g+C$
Arcsine reverse chain
$\int\frac{g'}{\sqrt{1-g^2}}\,dx=\arcsin g+C$
Product rule differential form
$d(uv)=u\,dv+v\,du$
Integration by parts
$\int u\,dv=uv-\int v\,du$
LIATE heuristic
$\text{logarithmic}\to\text{inverse trig}\to\text{algebraic}\to\text{trig}\to\text{exponential}$
Logarithm antiderivative
$\int\ln x\,dx=x\ln x-x+C$
Arctangent antiderivative
$\int\arctan x\,dx=x\arctan x-\frac12\ln(1+x^2)+C$
Cyclic cosine result
$\int e^x\cos x\,dx=\frac12e^x(\sin x+\cos x)+C$
Cyclic sine result
$\int e^x\sin x\,dx=\frac12e^x(\sin x-\cos x)+C$
Definite integration by parts
$\int_a^b u\,dv=[uv]_a^b-\int_a^b v\,du$
Method-selection flow
$\text{simplify}\to\text{standard form}\to\text{substitution signal}\to\text{product signal}\to\text{verify}$
Substitution summary
$u=g(x),\quad du=g'(x)dx,\quad \int f(g(x))g'(x)dx=\int f(u)du$
Definite substitution summary
$\int_a^b f(g(x))g'(x)dx=\int_{g(a)}^{g(b)}f(u)du$
Parts summary
$\int u\,dv=uv-\int v\,du$
Definite parts summary
$\int_a^b u\,dv=[uv]_a^b-\int_a^b v\,du$
Topic 5.17Not started
Areas and volumes of revolution
Use definite integrals to find exact areas and volumes of revolution, choosing the correct orientation, limits, and disc or washer radii from a carefully sketched region.
245 min estimated
Topic 5.17 progress0%
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Learning objectives
Distinguish signed integral values from geometric area and split intervals where necessary.
Find areas between curves using vertical or horizontal slices.
Find intersection points and use them as justified limits of integration.
Model volumes formed by rotation about the x-axis or y-axis using discs and washers.
Identify outer and inner radii as perpendicular distances from the axis of rotation.
Use a GDC to locate intersections, evaluate definite integrals, and verify exact analytic work.
Present complete IB HL solutions with sketches, formulas, correct bounds, exact values, and units.
Key formulas
Fundamental Theorem of Calculus
$\int_a^b f(x)\,dx=F(b)-F(a),\qquad F'(x)=f(x)$
Reversing limits
$\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$
Total area
$A=\int_a^b|f(x)|\,dx$
Area under a non-negative curve
$A=\int_a^b f(x)\,dx\qquad\text{when }f(x)\ge0$
Area between curves with vertical slices
$A=\int_a^b\bigl(f(x)-g(x)\bigr)\,dx\qquad\text{when }f(x)\ge g(x)$
Intersection condition
$f(x)=g(x)$
Area with horizontal slices
$A=\int_c^d\bigl(R(y)-L(y)\bigr)\,dy$
Symmetric area
$A=2\int_0^a h(x)\,dx\qquad\text{for a symmetric region}$
Cross-sectional volume principle
$V=\int A_{\text{cross-section}}\,d\ell$
Disc method
$V=\pi\int_a^b [R(x)]^2\,dx$
Volume about the x-axis
$V=\pi\int_a^b [f(x)]^2\,dx$
Washer area
$A_{\text{washer}}=\pi(R^2-r^2)$
Washer volume about the x-axis
$V=\pi\int_a^b\left([R(x)]^2-[r(x)]^2\right)\,dx$
Disc volume about the y-axis
$V=\pi\int_c^d[R(y)]^2\,dy$
Washer volume about the y-axis
$V=\pi\int_c^d\left([R(y)]^2-[r(y)]^2\right)\,dy$
Distance from a horizontal axis
$\text{radius}=|f(x)-k|\qquad\text{about }y=k$
Distance from a vertical axis
$\text{radius}=|g(y)-h|\qquad\text{about }x=h$
Bounding-cylinder check
$0\le V\le\pi R_{\max}^2L$
Area between curves
$A=\int_a^b(\text{upper}-\text{lower})\,dx$
Area with horizontal slices
$A=\int_c^d(\text{right}-\text{left})\,dy$
Disc volume
$V=\pi\int R^2\,d\ell$
Washer volume
$V=\pi\int(R^2-r^2)\,d\ell$
Topic 5.18Not started
Differential equations
Model relationships between quantities and their rates of change using first-order differential equations, exact and numerical solution methods, equilibrium analysis, and real-world interpretation.
280 min estimated
Topic 5.18 progress0%
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Learning objectives
Interpret the order, variables and solution family of a differential equation.
Solve separable first-order differential equations and apply initial conditions.
Model exponential and logistic growth, decay and Newton cooling.
Identify equilibrium solutions and interpret direction fields and solution curves.
Approximate initial-value problems using Euler’s method.
Solve homogeneous first-order equations using the substitution y equals vx.
Solve linear first-order equations using an integrating factor.
Use a GDC appropriately and communicate domain restrictions, units and model limitations.
Key formulas
General first-order form
$\frac{dy}{dx}=f(x,y)$
General second-order form
$F\left(x,y,\frac{dy}{dx},\frac{d^2y}{dx^2}\right)=0$
Initial condition
$y(x_0)=y_0$
Separation rule
$\frac{dy}{dx}=g(x)h(y)\quad\Longrightarrow\quad\int\frac{1}{h(y)}\,dy=\int g(x)\,dx$
Equilibrium condition
$F(c)=0\quad\Longrightarrow\quad y=c$
Exponential model
$N(t)=N_0e^{kt}$
Doubling time
$T_2=\frac{\ln2}{k}\quad(k>0)$
Half-life
$T_{\frac{1}{2}}=\frac{\ln2}{|k|}\quad(k<0)$
Newton cooling solution
$T(t)=T_a+(T_0-T_a)e^{-kt}$
Logistic differential equation
$\frac{dN}{dt}=kN(M-N)$
Logistic solution
$N(t)=\frac{M}{1+Ae^{-kMt}}$
Logistic partial fractions
$\frac{1}{N(M-N)}=\frac1M\left(\frac1N+\frac1{M-N}\right)$
Euler update
$x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h f(x_n,y_n)$
Euler local line
$y\approx y_n+f(x_n,y_n)(x-x_n)\qquad(x_n\le x\le x_n+h)$
Homogeneous substitution
$y=vx\quad\Longrightarrow\quad\frac{dy}{dx}=v+x\frac{dv}{dx}$
Linear standard form
$y'+P(x)y=Q(x)$
Integrating factor
$\mu(x)=e^{\int P(x)\,dx}$
Linear solution structure
$y=\frac{1}{\mu(x)}\left(\int\mu(x)Q(x)\,dx+C\right)$
Product derivative match
$(\mu y)'=\mu y'+\mu'y,\qquad \frac{\mu'}{\mu}=P$
Separable equation
$\frac{dy}{dx}=g(x)h(y)\quad\Longrightarrow\quad\int\frac{1}{h(y)}\,dy=\int g(x)\,dx$
Homogeneous substitution
$y=vx,\qquad\frac{dy}{dx}=v+x\frac{dv}{dx}$
Integrating factor
$\mu=e^{\int P(x)\,dx}\quad\text{for}\quad y'+P(x)y=Q(x)$
Linear solution
$y=\frac{1}{\mu}\left(\int\mu Q\,dx+C\right)$
Topic 5.19Not started
Maclaurin series
Represent functions near the origin with Maclaurin series, generate new expansions algebraically or from differential equations, and use truncated series for approximation and limit evaluation.
290 min estimated
Topic 5.19 progress0%
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Learning objectives
Interpret power series and distinguish Taylor series from Maclaurin series.
Find Maclaurin coefficients from derivatives evaluated at zero.
Recall the required exponential, trigonometric, logarithmic, arctangent and binomial expansions.
Generate new expansions by substitution, addition, multiplication, differentiation and integration.
Develop a Maclaurin series from a differential equation and initial data.
Use truncated series to approximate function values and describe the order of approximation.
Evaluate limits by identifying the first surviving nonzero term.
State intervals of validity and communicate IB HL solutions with appropriate accuracy and justification.
Key formulas
Power series about a
$\sum\limits_{n=0}^{\infty}a_n(x-a)^n$
Power series about zero
$\sum\limits_{n=0}^{\infty}a_nx^n$
Taylor series
$f(x)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$
Maclaurin series
$f(x)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$
Coefficient formula
$a_n=\frac{f^{(n)}(0)}{n!}$
Truncated form
$f(x)=P_n(x)+O(x^{n+1})\quad\text{as }x\to0$
Exponential series
$e^x=\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$
Scaled exponential
$e^{kx}=1+kx+\frac{k^2x^2}{2!}+\frac{k^3x^3}{3!}+\cdots$
Sine series
$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$
Cosine series
$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$
Natural logarithm series
$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots$
Arctangent series
$\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots$
Generalized binomial series
$(1+x)^p=1+px+\frac{p(p-1)}{2!}x^2+\frac{p(p-1)(p-2)}{3!}x^3+\cdots$
General binomial coefficient
$\binom{p}{n}=\frac{p(p-1)\cdots(p-n+1)}{n!}$
Radius condition
$|x-a|<R$
Product coefficient
$c_n=\sum\limits_{k=0}^{n}a_kb_{n-k}$
Termwise differentiation
$\frac{d}{dx}\sum\limits_{n=0}^{\infty}a_nx^n=\sum\limits_{n=1}^{\infty}na_nx^{n-1}$
Termwise integration
$\int\sum\limits_{n=0}^{\infty}a_nx^n\,dx=C+\sum\limits_{n=0}^{\infty}\frac{a_n}{n+1}x^{n+1}$
Series assumption
$y=\sum\limits_{n=0}^{\infty}a_nx^n$
Shifted derivative
$y'=\sum\limits_{n=0}^{\infty}(n+1)a_{n+1}x^n$
Alternating-series error bound
$|R_n|\le\text{magnitude of the first omitted term}$
Leading-term ratio
$\frac{ax^m+O(x^{m+1})}{bx^m+O(x^{m+1})}\longrightarrow\frac ab\quad(x\to0)$
General Maclaurin formula
$f(x)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$
Exponential
$e^x=\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}$
Sine and cosine
$\sin x=x-\frac{x^3}{3!}+\cdots,\qquad\cos x=1-\frac{x^2}{2!}+\cdots$
Logarithm and arctangent
$\ln(1+x)=x-\frac{x^2}{2}+\cdots,\qquad\arctan x=x-\frac{x^3}{3}+\cdots$
Generalized binomial
$(1+x)^p=\sum\limits_{n=0}^{\infty}\binom pn x^n$
Series solution recurrence pattern
$y=\sum\limits_{n=0}^{\infty}a_nx^n,\qquad y'=\sum\limits_{n=0}^{\infty}(n+1)a_{n+1}x^n$

Mathematics: Analysis and Approaches

Course Overview

Continue where you left off

Course units

Number and Algebra

Topics in this unit

Scientific notation and operations with numbers

Learning objectives

Key formulas

Arithmetic sequences and series

Learning objectives

Key formulas

Geometric sequences and series

Learning objectives

Key formulas

Financial applications (current syllabus)

Learning objectives

Key formulas

Integer exponents and introduction to logarithms

Learning objectives

Key formulas

Simple deductive proof

Learning objectives

Key formulas

Rational exponents and logarithm laws

Learning objectives

Key formulas

Infinite geometric series

Learning objectives

Key formulas

Binomial theorem

Learning objectives

Key formulas

Counting principles and the extended binomial theorem

Learning objectives

Key formulas

Partial fractions

Learning objectives

Key formulas

Complex numbers in Cartesian form

Learning objectives

Key formulas

Polar, cis and exponential forms of complex numbers

Learning objectives

Key formulas

Conjugate roots and De Moivre’s theorem

Learning objectives

Key formulas

Proof by induction, contradiction and counterexample

Learning objectives

Key formulas

Systems of linear equations

Learning objectives

Key formulas

Functions

Topics in this unit

Equations of straight lines

Learning objectives

Key formulas

The concept of a function

Learning objectives

Key formulas

Graphs of functions

Learning objectives

Key formulas

Key features of graphs

Learning objectives

Key formulas

Composite and inverse functions

Learning objectives

Key formulas

Quadratic functions

Learning objectives

Key formulas

Quadratic equations and inequalities

Learning objectives

Key formulas

Reciprocal and rational functions

Learning objectives

Key formulas